📚 Maths Year 1 Pure: Question Type Breakdown | 数学第一年纯数题型解析
Year 1 Pure Mathematics lays the groundwork for the entire A level Maths course. A clear understanding of the standard question types not only saves time in the exam but also reveals the connections between topics. This revision guide sorts through the most common question formats – from simplifying algebraic expressions to integrating polynomials – and explains exactly what each one tests and how to tackle it effectively.
第一年纯数是整个 A level 数学的基础。清楚掌握常见题型不仅能在考场上节省时间，还能帮你发现不同主题之间的联系。这份复习指南梳理了最常见的出题模式——从化简代数表达式到多项式积分——详细解释每种题型考查的核心能力，并给出高效的解题思路。
1. Algebraic Expressions and Indices | 代数表达式与指数
Questions often start by asking you to simplify expressions involving negative and fractional indices. You must apply the laws of indices confidently: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ. A typical task is to rewrite a term like 5x⁻² as 5/x², then combine it with other terms into a single fraction.
这类题目通常要求化简含有负指数和分数指数的式子。你需要熟练运用指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ 以及 (aᵐ)ⁿ = aᵐⁿ。最常见的操作是把 5x⁻² 写成 5/x²，再与其他项合并为单一分式。
Another frequent format presents a surd or root expression, expecting you to rewrite it using fractional powers. For instance, √x can be expressed as x½, and ∛(x²) as x⅔. Then you apply the same index laws to simplify or differentiate later.
另一种高频题型是给出根式，要求用分数指数表示。例如 √x 写成 x½，∛(x²) 写成 x⅔。接着就用同样的指数定律进行化简，或为后续的微积分步骤做准备。
2. Quadratics and Completing the Square | 二次函数与配方法
The three forms you must switch between are expanded form (ax² + bx + c), factorised form ((px + q)(rx + s)) and completed square form (a(x + p)² + q). Questions frequently ask you to complete the square for a quadratic, then state the coordinates of the vertex or solve an equation.
你必须能自由切换二次函数的三种形式：展开式 (ax² + bx + c)、因式分解式 ((px + q)(rx + s)) 和配方式 (a(x + p)² + q)。题目经常要求对二次式配方，然后写出顶点坐标或解方程。
With the completed square form you can also determine the discriminant Δ = b² − 4ac and interpret it: two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0. Exam questions often combine this with sketching the graph, labelling the vertex and intercepts.
利用配方式你还能得出判别式 Δ = b² − 4ac 并解释其意义：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 没有实根。考试常将这一点与画图结合，要求标出顶点和截距。
3. Equations and Inequalities | 方程与不等式
Linear equations are straightforward, but watch for those that involve algebraic fractions. The standard method is to multiply through by the common denominator, then solve the resulting linear equation. Always check for extraneous solutions if the denominator contains the variable.
一次方程虽简单，但要留意涉及代数分式的题型。标准方法是两边同乘公分母，然后解所得的一次方程。如果分母含变量，必须检验是否会引入增根。
Quadratic inequalities require a careful sign analysis. After rearranging to one side and factorising, you draw a sketch or a sign table to determine the intervals where the inequality holds. Solutions are typically expressed as {x : x < a} ∪ {x : x > b} or using interval notation.
二次不等式需要仔细的符号分析。先移项并因式分解，然后通过画草图或列符号表找出使不等式成立的区间。解集常写成 {x : x < a} ∪ {x : x > b} 或用区间记号表示。
4. Graphs and Transformations | 图像与变换
You are expected to know the shapes of basic curves: y = x², y = x³, y = 1/x, y = √x, y = 1/x² etc. Then transformations are applied: f(x) + a is a vertical translation, f(x + a) a horizontal translation, a f(x) a vertical stretch, f(ax) a horizontal stretch. Questions give the transformed equation and ask for the new asymptotes or intercepts.
你需要熟悉基本曲线的形状：y = x², y = x³, y = 1/x, y = √x, y = 1/x² 等。然后叠加变换：f(x) + a 是竖直平移，f(x + a) 是水平平移，a f(x) 是竖直拉伸，f(ax) 是水平拉伸。题目常给出变换后的方程，要求找出新渐近线或截距。
A common pitfall is the order of transformations. Typically you apply horizontal changes first, or work ‘inside the bracket’ before ‘outside’. Exam questions often test this by asking you to describe the sequence of transformations that maps one function onto another.
常见陷阱是变换顺序。通常应先处理水平方向的改变，也就是‘括号内’的变换先于‘括号外’的变换。考试常通过要求描述从原函数到新函数的变换序列来考查这一点。
5. Straight Line Graphs | 直线图像
The equation of a straight line can appear as y = mx + c, ax + by + c = 0 or y – y₁ = m(x – x₁). Questions ask you to find the gradient from two points, then write the equation in a specific format. Parallel lines share the same gradient; perpendicular lines satisfy m₁ × m₂ = –1.
直线方程可以写成 y = mx + c、ax + by + c = 0 或 y – y₁ = m(x – x₁) 的形式。题目可能要求通过两点求斜率，再写出特定形式的方程。平行线斜率相等；垂直线满足 m₁ × m₂ = –1。
Modelling questions use straight lines to represent real‑world situations: for example, cost against number of units. You need to interpret the gradient and intercept in context, and use the equation to make predictions.
建模题用直线表示现实情境，例如成本随产量变化。你要结合背景解释斜率和截距的含义，并利用方程做出预测。
6. Circles | 圆的方程
The standard circle equation (x – a)² + (y – b)² = r² gives the centre (a, b) and radius r directly. A favourite exam question presents the circle in expanded form x² + y² + 2gx + 2fy + c = 0 and asks you to complete the square to find the centre and radius.
标准圆方程 (x – a)² + (y – b)² = r² 直接给出圆心 (a, b) 和半径 r。考试喜欢给出展开式 x² + y² + 2gx + 2fy + c = 0，要求你通过配方找出圆心和半径。
Tangent and chord questions appear regularly. The key property is that the radius to the point of tangency is perpendicular to the tangent. You use this to find the equation of a tangent, often after establishing the gradient of the radius with two points.
切线与弦的题目经常出现。核心性质是半径与切点在切线处垂直。你通常需要先算出半径所在直线的斜率，再利用垂直关系求出切线方程。
7. Algebraic Division and Factor Theorem | 代数除法与因式定理
Polynomial division can be done by long division or by comparing coefficients. The factor theorem states that if f(a) = 0, then (x – a) is a factor of f(x). Typical questions give a cubic and one known factor, then ask you to fully factorise it.
多项式除法可以用长除法或比较系数法。因式定理指出，若 f(a) = 0，则 (x – a) 是 f(x) 的因式。典型题目给一个三次式和一个已知因式，要求完全因式分解。
The remainder theorem is another variation: after dividing f(x) by (x – a), the remainder equals f(a). This is used to find unknown coefficients in a polynomial by setting up equations from given remainders.
余数定理是另一种变体：f(x) 除以 (x – a) 的余式等于 f(a)。常用来求多项式中的未知系数，即根据给定的余数值建立方程求解。
8. Binomial Expansion | 二项式展开
For a positive integer index n, the expansion of (a + b)ⁿ follows Pascal’s triangle or the formula with nCr coefficients. Questions ask you to expand expressions like (2 + 3x)⁴, giving terms in ascending powers of x. Often you then use the expansion to estimate a value, such as (2.003)⁴.
对于正整数指数 n，(a + b)ⁿ 的展开遵循帕斯卡三角形或带有 nCr 系数的公式。题目要求展开像 (2 + 3x)⁴ 这样的式子，并写出 x 的升幂排列项。然后常利用展开式估算如 (2.003)⁴ 这样的数值。
When there are two brackets, a common task is to multiply the first few terms of expansions to find the coefficient of a specific power, without fully expanding everything. This requires careful selection of terms whose products give the desired xᵏ.
如果涉及两个括号，常见做法是将展开式的前几项相乘，直接找出特定次幂的系数，而不必展开整个式子。这需要精心挑选那些乘积能得到目标 xᵏ 的项。
9. Trigonometry | 三角学
The sine, cosine and tangent graphs are essential, as are exact values for angles such as 30°, 45° and 60°. Solving simple trigonometric equations like sin θ = 0.5 for 0° ≤ θ ≤ 360° involves using the CAST diagram or graph symmetry to find all solutions in the given interval.
正弦、余弦和正切函数的图像是基础，30°、45° 和 60° 等特殊角的准确值也必须熟记。解像 sin θ = 0.5（0° ≤ θ ≤ 360°）这样简单的三角方程，需要借助 CAST 图或图像对称性找出给定区间内的全部解。
Identities such as tan θ ≡ sin θ / cos θ and sin²θ + cos²θ ≡ 1 are tested by asking you to simplify an expression or prove a given identity. The strategy is usually to rewrite everything in terms of sine and cosine, then apply the Pythagorean identity.
恒等式如 tan θ ≡ sin θ / cos θ 和 sin²θ + cos²θ ≡ 1 常以化简或证明题的形式出现。解题策略通常是将式子全用正弦和余弦表示，再运用平方恒等式。
10. Vectors | 向量
Vector questions often supply coordinates of points and ask for the vector joining them, its magnitude and possibly a unit vector in the same direction. The notation a = xi + yj is standard, and you use Pythagoras to find the magnitude |a| = √(x² + y²).
向量题常给出一组点的坐标，要求写出连接两点的向量、其模长，有时还要求同方向的单位向量。记法 a = xi + yj 是标准形式，模长用勾股定理计算：|a| = √(x² + y²)。
Geometric problems involve parallel vectors (scalar multiples), position vectors and the section formula. The real test is whether you can translate a geometric relationship, such as a point dividing a line segment in a given ratio, into a vector equation.
几何问题涉及平行向量（标量倍数）、位置向量和定比分点公式。真正的考验在于能否把几何关系——例如点按给定比例分割线段——转化为向量方程。
11. Differentiation | 微分
The core routine is to differentiate powers of x: if y = xⁿ then dy/dx = nxⁿ⁻¹. This extends to sums and constant multiples. Questions often start with a non-simplified expression, requiring you to rewrite terms with negative or fractional indices before differentiating.
微分的核心操作是对 x 的幂函数求导：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。这可以推广到多项式和常数倍。题目给出的常常是未化简的式子，你必须先用负指数或分数指数重写，再逐项求导。
An important application is finding the gradient of a curve at a given point, which is dy/dx evaluated at that x-coordinate. Then you write the equation of the tangent or normal using the straight line formula. You also use the sign of dy/dx to determine where a function is increasing or decreasing.
一个重要应用是求曲线上某点的切线斜率，即把该点 x 坐标代入 dy/dx。然后使用直线方程写出切线或法线方程。你还可以根据 dy/dx 的正负判断函数的递增或递减区间。
12. Integration | 积分
Integration is treated as the reverse of differentiation. The rule ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, for n ≠ –1, is applied term by term. Questions typically start with a derivative-like expression and ask for the indefinite integral, so you must first rewrite the integrand in power form.
积分被视作微分的逆运算。公式 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c（n ≠ –1）需逐项使用。题目通常给一个类似导数的式子，要求计算不定积分，因此你首先要把被积函数写成幂函数的形式。
Definite integration is used to calculate the area under a curve between two x-values. The method is to integrate, substitute the upper and lower limits, and subtract. A common trick is the area between two curves, where you subtract the lower curve’s y-value from the upper one before integrating.
定积分用于计算曲线与 x 轴之间的面积。方法是积分后代入上限和下限相减。有一个常见技巧是求两曲线之间的面积，此时要先从上方曲线的 y 值中减去下方曲线的 y 值，再对差值积分。
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