📚 AQA International A-level Pure Maths P2: Core Topics and Exam Techniques | AQA 国际 A-level 纯数学 P2：核心专题与应试技巧
The AQA International A-level Pure Mathematics 2 (P2) unit builds on P1 and deepens your understanding of proof, algebra, functions, coordinate geometry, sequences, trigonometry, exponentials, logarithms and calculus. This revision guide summarises the essential knowledge, the standard formulas and the common exam skills required for each topic.
AQA 国际 A-level 纯数学 2（P2）单元在 P1 的基础上深化你对证明、代数、函数、坐标几何、数列、三角学、指数、对数以及微积分的理解。本复习指南总结了每个专题的核心知识、标准公式与常见考试技巧。
1. Algebraic Methods and Proof | 代数方法与证明
P2 extends algebraic manipulation to algebraic division, the factor theorem, the remainder theorem and proof by contradiction. You must be able to factorise cubic and quartic expressions after identifying a factor.
P2 将代数运算扩展到代数除法、因式定理、余式定理和反证法。你必须能够在确定一个因式后对三次和四次表达式进行因式分解。
a factor, then f(a) = 0. This is the tool examiners expect you to use when a question says "show that (x − 3) is a factor" or "hence factorise fully".
因式定理指出，如果多项式 f(x) 满足 f(a) = 0，则 (x − a) 是 f(x) 的一个因式。反过来，如果 (x − a) 是因式，则 f(a) = 0。当题目要求"证明 (x − 3) 是因式"或"由此完全因式分解"时，这就是考官期望你使用的方法。
f(a) = 0 ⇔ (x − a) is a factor of f(x)
The remainder theorem gives the remainder when f(x) is divided by (x − a) as f(a). If the divisor is (bx − a), the remainder is f(a ÷ b) instead. A zero remainder means the divisor is a factor.
余式定理给出 f(x) 除以 (x − a) 的余数为 f(a)。如果除式是 (bx − a)，余数则改为 f(a ÷ b)。余数为零意味着该除式是一个因式。
uch as 1 = 0 or an integer being both odd and even. Classic P2 proofs include showing that √2 is irrational and that there are infinitely many prime numbers.
反证法要求你假设命题的反面成立，并证明这会导致逻辑矛盾，例如 1 = 0 或某个整数既是奇数又是偶数。P2 的经典证明题包括证明 √2 是无理数，以及素数有无穷多个。
When dividing polynomials, write missing terms with zero coefficients, for example x³ + 0x² + 0x − 8, to avoid errors in long division. Always line up powers of x in descending order before you start.
进行多项式除法时，将缺失项写成零系数，例如 x³ + 0x² + 0x − 8，以避免长除法中的错误。开始之前，务必按 x 的降幂对齐各项。
2. Functions, Domain, Range and Inverses | 函数、定义域、值域与反函数
You need to understand domain, range, one-to-one and many-to-one functions, inverse functions, and how to combine functions into composites such as fg(x).
你需要理解定义域、值域、一一函数和多对一函数、反函数，以及如何将函数组合成复合函数，例如 fg(x)。
A function is one-to-one if every output value comes from exactly one input value. Only one-to-one functions have inverses, because f⁻¹ must itself be a function. Many-to-one functions can be made invertible by restricting the domain.
如果每个输出值都恰好来自一个输入值，该函数就是一一函数。只有一一函数才有反函数，因为 f⁻¹ 本身必须是一个函数。多对一函数可以通过限制定义域变为可逆的。
f⁻¹. To find an inverse algebraically, write y = f(x), rearrange to make x the subject, then swap x and y.
y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的镜像，且 f 的定义域变成 f⁻¹ 的值域，f 的值域变成 f⁻¹ 的定义域。用代数方法求反函数时，令 y = f(x)，整理使 x 成为主元，然后交换 x 与 y。
e range with the correct inequality sign, using ≥ or > rather than a vague description.
二次函数的值域可通过配方确定顶点来求得。例如 y = (x − 2)² + 5 的值域是 y ≥ 5。陈述值域时必须使用正确的不等号，使用 ≥ 或 >，而不是含糊的描述。
Composite functions require the range of the inner function to lie inside the domain of the outer function. If it does not, the composite is not defined for those values, and this restriction must be stated.
复合函数要求内层函数的值域落在外层函数的定义域之内。如果不满足，则该复合函数对这些值无定义，必须说明这一限制。
3. Graph Transformations and Modulus | 图像变换与绝对值
ections of the graph of y = f(x), and to write down the equation of the transformed curve.
你必须能够画出并描述 y = f(x) 图像的平移、伸缩和翻转，并写出变换后曲线的方程。
| Transformation | Equation | Effect |
| Translation | y = f(x − a) | Shift a units right |
| Translation | y = f(x) + b | Shift b units up |
| Stretch | y = af(x) | Vertical stretch, factor a |
| Stretch | y = f(x ÷ a) | Horizontal stretch, factor a |
| Reflection | y = −f(x) | Reflect in the x-axis |
| Reflection | y = f(−x) | Reflect in the y-axis |
Modulus functions such as y = |x| and y = |f(x)| se linear or quadratic branches before solving equations or inequalities. The graph of y = |f(x)| d by reflecting any part of y = f(x) that lies below the x-axis into the upper half-plane.
形如 y = |x| 和 y = |f(x)| 的绝对值函数通常需要拆分为分段线性或二次分支，然后再求解方程或不等式。y = |f(x)| 的图像是将 y = f(x) 位于 x 轴下方的部分翻折到上半平面得到的。
For an equation such as |2x − 5| the original equation. For inequalities, sketch both sides and read off the intersection points to identify the solution set.
对于 |2x − 5| = 7 这样的方程，你必须同时求解 2x − 5 = 7 和 2x − 5 = −7，然后把每个解代回原方程检验。对于不等式，应画出两边图像并读出交点，以确定解集。
4. Coordinate Geometry: Circles and Lines | 坐标几何：圆与直线
P2 coordinate geometry focuses on the equation of a circle, tangents and normals, and the link between algebra and geometry through completing the square.
P2 的坐标几何重点是圆的方程、切线与法线，以及通过配方建立的代数与几何之间的联系。
(x − a)² + (y − b)² = r²
y − 12 = 0, complete the square in x and in y separately to recover the centre and radius.
圆心为 (a, b)、半径为 r 的圆具有上述方程。如果给出的方程是展开形式，例如 x² + y² − 6x + 4y − 12 = 0，则分别对 x 和 y 配方，以还原圆心与半径。
A tangent to a circle is perpendicular to the radius at the point of contact, so the product of the two gradients is −1. The normal at a point on the circle is simply the radius produced, so it passes through the centre.
圆的切线与接触点处的半径垂直，因此两条直线的斜率之积为 −1。圆上某点的法线就是该半径的延长线，因此它经过圆心。
To decide whether a line intersects, touches or misses a circle, substitute the line equation into the circle equation and examine the discriminant. Two roots mean two intersections, one repeated root means the line is a tangent, and no real roots means the line misses the circle.
要判断一条直线与圆相交、相切还是不相交，可将直线方程代入圆方程并考察判别式。两个根表示两个交点，一个重根表示直线相切，无实根表示直线与圆不相交。
int formula, the gradient formula and the condition for perpendicular lines, m₁ × m₂ = −1.
你应熟练使用的标准工具包括距离公式、中点公式、斜率公式，以及垂直条件 m₁ × m₂ = −1。
5. Sequences and Series | 数列与级数
You must recognise and work with arithmetic sequences, geometric sequences, sigma notation and the binomial expansion for positive integer powers.
你必须能够识别并处理等差数列、等比数列、求和符号，以及正整数次幂的二项式展开。
uₙ = a + (n − 1)d, Sₙ = ½n[2a + (n − 1)d]
first n terms is Sₙ = a(1 − rⁿ) ÷ (1 − r) for r ≠ 1. The sum to infinity exists only when |r| < 1 and equals 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)（r ≠ 1）。仅当 |r| < 1 时无穷级数收敛，其和为 a ÷ (1 − r)。
The binomial expansion of (a + b)ⁿ for a positive integer n uses binomial coefficients, which may be written as nCr or in the notation of the formula booklet. Remember that the powers of a decrease while the powers of b increase, and that the sum of the two indices is always n.
正整数 n 的 (a + b)ⁿ 二项式展开使用二项式系数，可写作 nCr 或公式手册中的记法。记住 a 的幂递减而 b 的幂递增，且两个指数之和始终等于 n。
Common exam tasks include finding a particular coefficient, finding the term independent of x, and using the expansion to approximate a value such as 1.02¹⁰. For approximations, state the range of validity of the expansion.
常见考题包括求某一特定系数、求不含 x 的项，以及利用展开式近似计算如 1.02¹⁰ 的值。做近似计算时，要说明展开式的有效范围。
6. Trigonometry: Identities and Equations | 三角学：恒等式与方程
P2 trigonometry works in radians as well as degrees. You must be able to convert between them and to use the exact values of sine, cosine and tangent at 0, π÷6, π÷4, π÷3 and π÷2.
P2 的三角学同时使用弧度与角度。你必须能够在两者之间换算，并会使用正弦、余弦、正切在 0、π÷6、π÷4、π÷3 和 π÷2 处的精确值。
arc length = rθ, sector area = ½r²θ
, converts most quadratic trigonometric equations into a form you can factorise. If an equation contains both sin²θ and cos θ, replace sin²θ with 1 − cos²θ first.
基本恒等式 sin²θ + cos²θ = 1，连同 tan θ = sin θ ÷ cos θ，可将大多数二次三角方程转化为可因式分解的形式。如果方程中同时含有 sin²θ 和 cos θ，应先用 1 − cos²θ 替换 sin²θ。
equation in 2θ to one in θ. Choose the version of cos 2θ that matches the other terms in the equation.
倍角公式 sin 2θ = 2 sin θ cos θ 以及 cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1，可将关于 2θ 的方程化为关于 θ 的方程。应选择与方程中其他项相匹配的 cos 2θ 形式。
ery solution inside the interval and no others. Sketch the graph of the trigonometric function to confirm how many solutions exist before you start.
在求解给定区间（如 0 ≤ θ ≤ 2π）内的方程时，必须列出区间内的全部解且不多列。开始之前，先画出该三角函数的图像，以确认解的个数。
ns of the form a sin θ + b cos θ can be written as R sin(θ ± α) or R cos(θ − α), where R = √a² + b². This is essential for finding maximum and minimum values and for solving equations that cannot be factorised.
形如 a sin θ + b cos θ 的表达式可写成 R sin(θ ± α) 或 R cos(θ − α)，其中 R = √a² + b²。这在求最大值、最小值，以及求解无法因式分解的方程时至关重要。
7. Exponentials and Logarithms | 指数与对数
You need to know the shape and properties of y = eˣ and y = ln x, the laws of logarithms, and how to solve equations of the form aˣ = b.
你需要了解 y = eˣ 与 y = ln x 的图像形状和性质、对数运算法则，以及如何求解形如 aˣ = b 的方程。
log(ab) = log a + log b, log(a ÷ b) = log a − log b, log(aⁿ) = n log a
To solve aˣ = b, take logarithms of both sides and use the power law to bring x down. For example, 3ˣ = 20 gives x = log 20 ÷ log 3. If the base is e, use ln directly.
要解 aˣ = b，可对两边取对数并用幂法则把 x 降下来。例如 3ˣ = 20 给出 x = log 20 ÷ log 3。如果底数为 e，则直接使用 ln。
the constants: in y = Aeᵏᵗ, A is the initial value and k is the continuous rate of change.
指数模型描述增长与衰减，例如人口增长或放射性衰变。关键考试技巧是解释常数的含义：在 y = Aeᵏᵗ 中，A 是初始值，k 是连续变化率。
y against log x is a straight line with gradient n and intercept log a. Similarly, y = abˣ becomes log y = log a + x log b. This "reduction to linear form" question appears regularly.
如果关系式可写成 y = axⁿ，取对数得 log y = log a + n log x，因此以 log y 对 log x 作图是一条直线，斜率为 n，截距为 log a。类似地，y = abˣ 变为 log y = log a + x log b。这种"化为线性形式"的题目经常出现。
8. Differentiation Techniques | 微分技巧
Beyond the basic rule for polynomials, P2 requires the chain rule, the product rule and the quotient rule, together with the derivatives of exponential, logarithmic and trigonometric functions.
除多项式的基本求导法则外，P2 还要求掌握链式法则、乘积法则和商法则，以及指数函数、对数函数和三角函数的导数。
| Function | Derivative |
| eˣ | eˣ |
| ln x | 1 ÷ x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| sin ax | a cos ax |
The quotient rule states that if y = u ÷ v then dy ÷ dx = [v(du ÷ dx) − u(dv ÷ dx)] ÷ v².
链式法则用于求复合函数的导数：dy ÷ dx = dy ÷ du × du ÷ dx。乘积法则指出，若 y = uv，则 dy ÷ dx = u(dv ÷ dx) + v(du ÷ dx)。商法则指出，若 y = u ÷ v，则 dy ÷ dx = [v(du ÷ dx) − u(dv ÷ dx)] ÷ v²。
The second derivative d²y ÷ dx² identifies the nature of a stationary point: if it is negative the point is a maximum, if positive a minimum. You should also be able to determine where a function is increasing or decreasing by examining the sign of the first derivative.
二阶导数 d²y ÷ dx² 可判断驻点的性质：若为负则是极大值点，若为正则是极小值点。你还应能通过考察一阶导数的符号，判断函数在何处递增或递减。
Implicit differentiation treats y as a function of x, so differentiating y² gives 2y(dy ÷ dx).
在应用题中，链式法则将相关变化率联系起来，例如 dV ÷ dt = (dV ÷ dr) × (dr ÷ dt)。隐函数求导把 y 视为 x 的函数，因此对 y² 求导得到 2y(dy ÷ dx)。
9. Integration Techniques | 积分技巧
P2 integration includes the reverse of the standard derivatives, definite integrals, areas under curves and between curves, and simple substitution or integration by parts.
P2 的积分包括标准导数公式的逆用、定积分、曲线下面积与曲线间面积，以及简单的换元积分或分部积分。
∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + c for n ≠ −1, ∫ (1 ÷ x) dx = ln|x| + c
derivative of the inner function appears as a factor. If it does not appear, either use a formal substitution or adjust the constant carefully.
反向链式法则可处理如 ∫ 2x(x² + 1)⁵ dx 之类的积分，其中内层函数的导数作为因式出现。若不出现，则需使用正式的换元法，或小心地调整常数。
Integration by parts follows the formula ∫ u(dv ÷ dx) dx = uv − ∫ v(du ÷ dx)
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图