Edexcel A-Level Maths: Core Pure Revision | 爱德思A-Level数学：核心纯数学复习

📚 Edexcel A-Level Maths: Core Pure Revision | 爱德思A-Level数学：核心纯数学复习

This revision guide brings together the most common pure mathematics skills tested in Edexcel A-Level Mathematics. It is designed to help you consolidate algebraic techniques, coordinate geometry, trigonometry, calculus, sequences and vectors in one structured review.

本复习指南汇集了爱德思A-Level数学纯数学部分最常见的核心技能。通过结构化梳理，帮助你巩固代数运算、坐标几何、三角学、微积分、数列和向量等关键考点。

1. Algebraic Manipulation and Proof | 代数运算与证明

Algebraic fluency is the foundation of the A-Level pure mathematics course. You must be able to factorise polynomials, simplify rational expressions and construct clear mathematical proofs using deduction, exhaustion or contradiction.

代数运算能力是A-Level纯数学课程的基础。你需要掌握因式分解、化简有理式，并能够使用演绎、穷举或反证法构建清晰的数学证明。

A key tool is the factor theorem: if f(a) = 0 for a polynomial f(x), then (x – a) is a factor. This allows you to break down cubic and quartic expressions before solving equations or sketching graphs.

因式定理是一个关键工具：若多项式 f(x) 满足 f(a) = 0，则 (x – a) 是其因式。这帮助你在解方程或画图前分解三次、四次多项式。

Proof questions often ask you to show that a statement holds for all integers or all positive values. Common approaches include algebraic rearrangement, squaring both sides for inequalities only when both sides are non-negative, and exhausting a small set of possible cases.

证明题常要求证明某个结论对所有整数或所有正数成立。常见方法包括代数变形、在两边非负时对不等式两边平方，以及对少量可能情况进行穷举。

2. Quadratics and Inequalities | 二次函数与不等式

Quadratics appear throughout the A-Level course, from completing the square and finding the vertex to using the discriminant to determine the number of roots. The discriminant Δ = b² – 4ac tells you whether a quadratic has two distinct real roots, one repeated root, or no real roots.

二次函数贯穿A-Level课程，从配方法和求顶点，到利用判别式判断根的个数。判别式 Δ = b² – 4ac 可以判断二次方程有两个不同实根、一个重根还是没有实根。

When solving quadratic inequalities such as ax² + bx + c > 0, sketch the graph or use a sign diagram after finding the roots. This helps you avoid common sign errors when the coefficient of x² is negative.

在解 ax² + bx + c > 0 这类二次不等式时，建议先求根，再画草图或使用符号表。这样当 x² 项系数为负时，可以避免常见的符号错误。

Completing the square is also the quickest way to find the minimum or maximum value of a quadratic expression. For example, x² + 6x + 2 = (x + 3)² – 7, so the minimum value is -7 at x = -3.

配方法也是求二次表达式最小值或最大值的最快方法。例如 x² + 6x + 2 = (x + 3)² – 7，因此最小值为 -7，对应 x = -3。

3. Polynomials and the Factor Theorem | 多项式与因式定理

Polynomial division is needed when the divisor is not linear, or when finding a quotient and remainder. The remainder theorem states that when f(x) is divided by (x – a), the remainder is f(a).

当除数不是一次式，或需要求商式和余数时，需要使用多项式除法。余数定理指出，当 f(x) 除以 (x – a) 时，余数为 f(a)。

To factorise a cubic such as f(x) = x³ – 4x² + x + 6, test small integer values of x. Since f(-1) = 0, (x + 1) is a factor; then divide to get a quadratic and factorise further if possible.

要分解三次多项式，例如 f(x) = x³ – 4x² + x + 6，先代入小整数检验。因为 f(-1) = 0，所以 (x + 1) 是因式；再进行除法得到二次式，并尽可能继续分解。

Sketching polynomial graphs requires knowledge of the sign of the leading coefficient, repeated roots and the behaviour at x → ±∞. A repeated root touches the x-axis without crossing it.

绘制多项式图像需要了解首项系数的符号、重根以及 x → ±∞ 时的变化趋势。重根处图像与x轴相切但不穿越。

4. Coordinate Geometry and Circles | 坐标几何与圆

The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r². If the equation is given in expanded form, complete the square for both x and y to find the centre and radius.

圆心为 (a, b)、半径为 r 的圆的方程为 (x – a)² + (y – b)² = r²。如果给出的是展开式，需分别对 x 和 y 进行配方法，以求出圆心和半径。

Tangents to a circle are perpendicular to the radius at the point of contact. To find a tangent equation, first find the gradient of the radius, then use the negative reciprocal gradient and the tangent point.

圆的切线在切点处与半径垂直。求切线方程时，先求半径的斜率，然后取负倒数作为切线斜率，并利用切点代入点斜式。

When solving line and circle intersection problems, substitute y = mx + c into the circle equation and solve the resulting quadratic. If the discriminant is zero, the line is a tangent; if positive, it cuts the circle at two points.

解直线与圆的交点问题时，将 y = mx + c 代入圆方程并求解所得二次方程。若判别式为零，直线为切线；若为正，则直线与圆有两个交点。

5. Trigonometry: Identities and Equations | 三角恒等式与方程

Exact trigonometric values such as sin 30° = ½, cos 45° = √2/2 and tan 60° = √3 are frequently required without a calculator. You should know these from the special triangles or from the unit circle.

常见精确三角值，如 sin 30° = ½、cos 45° = √2/2、tan 60° = √3，经常要求在不使用计算器的情况下给出。你需要通过特殊三角形或单位圆熟记这些值。

The Pythagorean identity sin² θ + cos² θ = 1 and the double angle formulas sin 2θ = 2 sin θ cos θ, cos 2θ = cos² θ – sin² θ are essential for solving equations and proving identities.

勾股恒等式 sin² θ + cos² θ = 1 以及倍角公式 sin 2θ = 2 sin θ cos θ、cos 2θ = cos² θ – sin² θ 是解方程和证明恒等式的重要工具。

When solving trigonometric equations, state the interval clearly, find the principal value using inverse functions, and then use symmetry or periodicity to find all solutions. Working in radians is often required in pure mathematics.

解三角方程时，要明确给定区间，利用反三角函数求主值，然后通过对称性或周期性找到所有解。纯数学部分常要求使用弧度制。

Identity	Use
tan θ = sin θ / cos θ	Converting between tan and sin/cos forms
sin² θ = 1 – cos² θ	Simplifying expressions with only one trig function
cos 2θ = 2 cos² θ – 1	Reducing powers and solving double angle equations

6. Exponentials and Logarithms | 指数与对数

The natural exponential function eˣ and the natural logarithm ln x are inverse functions. This means ln(eˣ) = x and e^(ln x) = x for valid domains.

自然指数函数 eˣ 与自然对数 ln x 互为反函数。这意味着在有效定义域内有 ln(eˣ) = x 以及 e^(ln x) = x。

Key laws of logarithms include log(xy) = log x + log y, log(x/y) = log x – log y and log(xⁿ) = n log x. These laws apply to any base, including e and base 10.

对数的关键法则包括 log(xy) = log x + log y、log(x/y) = log x – log y 以及 log(xⁿ) = n log x。这些法则适用于任何底数，包括 e 和 10。

Exponential growth and decay models take the form P = P₀e^(kt), where k > 0 gives growth and k < 0 gives decay. You may need to use logarithms to find unknown constants or times.

指数增长与衰减模型通常写作 P = P₀e^(kt)，其中 k > 0 表示增长，k < 0 表示衰减。你常常需要使用对数求未知常数或时间。

7. Differentiation: Rules and Applications | 微分：法则与应用

Differentiation from first principles uses the limit of a difference quotient. For y = f(x), the derivative is f'(x) = lim_(h→0) [f(x+h) – f(x)] / h. Edexcel often tests this for simple polynomials.

从第一性原理求导使用差商的极限。对于 y = f(x)，导数为 f'(x) = lim_(h→0) [f(x+h) – f(x)] / h。爱德思常对简单多项式考查这一推导过程。

The main rules are the chain rule, product rule and quotient rule. For y = (u(x))ⁿ, dy/dx = n uⁿ⁻¹ du/dx; for y = uv, dy/dx = u’v + uv’; for y = u/v, dy/dx = (u’v – uv’) / v².

主要求导法则包括链式法则、乘法法则和除法法则。对 y = (u(x))ⁿ，dy/dx = n uⁿ⁻¹ du/dx；对 y = uv，dy/dx = u’v + uv’；对 y = u/v，dy/dx = (u’v – uv’) / v²。

Derivatives are used to find stationary points, classify maxima and minima through the second derivative or sign change, and determine gradients of tangents and normals to curves.

导数用于求驻点，通过二阶导数或导数符号变化判断极大值与极小值，以及求曲线切线和法线的斜率。

d/dx [sin x] = cos x, d/dx [eˣ] = eˣ, d/dx [ln x] = 1/x

8. Integration: Techniques and Area | 积分：技巧与面积

Integration reverses differentiation. The standard result ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C holds for n ≠ -1, and for exponentials, ∫ eˣ dx = eˣ + C.

积分是微分的逆运算。标准公式 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C 在 n ≠ -1 时成立；对指数函数有 ∫ eˣ dx = eˣ + C。

Integration by substitution is used when the integrand contains a function and its derivative. Let u = g(x), replace dx by du/g'(x), integrate with respect to u, then substitute back.

当被积函数包含一个函数及其导数时，常使用换元积分法。令 u = g(x)，将 dx 替换为 du/g'(x)，对 u 积分后再代回原变量。

Integration by parts is based on the product rule and is given by ∫ u dv = uv – ∫ v du. It is useful for products such as x eˣ or x sin x, where differentiation of one factor simplifies the integral.

分部积分法基于乘法法则，公式为 ∫ u dv = uv – ∫ v du。它适用于 x eˣ 或 x sin x 这类乘积，当其中一个因子求导后能使积分简化时非常有效。

Definite integrals give the exact area between a curve and the x-axis. If the curve is below the axis, the integral is negative and its absolute value is the area.

定积分给出曲线与x轴之间的精确面积。如果曲线位于x轴下方，积分值为负，其绝对值才是面积。

9. Sequences and Series | 数列与级数

An arithmetic sequence has a common difference d. Its nth term is aₙ = a + (n – 1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n – 1)d].

等差数列具有公差 d。第 n 项为 aₙ = a + (n – 1)d，前 n 项和为 Sₙ = n/2 [2a + (n – 1)d]。

A geometric sequence has a common ratio r. Its nth term is aₙ = arⁿ⁻¹, and the sum to n terms is Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1.

等比数列具有公比 r。第 n 项为 aₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 – rⁿ)/(1 – r)，其中 r ≠ 1。

When |r| < 1, an infinite geometric series converges, and its sum to infinity is S∞ = a/(1 – r). This is used in modelling and in recurring decimal problems.

当 |r| < 1 时，无穷等比级数收敛，其无穷和为 S∞ = a/(1 – r)。这一结果常用于建模和循环小数问题。

Sigma notation Σ means the sum of terms following a rule. Be careful with the lower and upper limits when evaluating sums by expanding or using standard formulas.

求和符号 Σ 表示按规则对各项求和。在通过展开或使用标准公式求值时，要特别注意求和下限和上限。

10. Vectors in 2D and 3D | 二维与三维向量

A vector has magnitude and direction. The magnitude of a vector a = xi + yj + zk is |a| = √(x² + y² + z²), and a unit vector in the same direction 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)，沿同方向的单位向量为 a/|a|。

The scalar product of two vectors a and b is a · b = |a||b| cos θ, where θ is the angle between them. In component form, a · b = x₁x₂ + y₁y₂ + z₁z₂.

两个向量的数量积为 a · b = |a||b| cos θ，其中 θ 为它们的夹角。在分量形式下，a · b = x₁x₂ + y₁y₂ + z₁z₂。

If a · b = 0 and both vectors are non-zero, the vectors are perpendicular. You can use this to prove right angles in geometric problems.

如果 a · b = 0 且两个向量均为非零向量，则它们互相垂直。你可以利用这一性质证明几何问题中的直角。

The vector equation of a line is r = a + λb, where a is a position vector and b is a direction vector. To show a point lies on a line, check whether its position vector satisfies the equation for some λ.

直线的向量方程为 r = a + λb，其中 a 是位置向量，b 是方向向量。要证明某点位于直线上，只需验证其位置向量能否对某个 λ 满足该方程。

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Edexcel A-Level Maths: Core Pure Revision | 爱德思A-Level数学：核心纯数学复习