📚 Common Maths Exam Question Patterns | A-level数学备考：常见考试题型归纳
In A-level mathematics, most papers test a surprisingly stable set of question types. Once you can recognise the underlying structure of each type, you can apply a clear method, avoid common pitfalls, and use exam time more efficiently.
在A-level数学中，大多数试卷考查的题型其实非常稳定。一旦你能够识别每类题目背后的结构，就能运用清晰的方法、避开常见陷阱，并在考试中更高效地利用时间。
1. Quadratic Functions and Equations | 二次函数与方程
Quadratics are usually the first major topic in pure mathematics. Exam questions often ask you to solve a quadratic equation, use the discriminant, or complete the square to find the turning point.
二次函数通常是纯数学部分的第一个大专题。考试题常要求你解二次方程、使用判别式，或通过配方法求顶点。
The most common command pattern is “find the range of values of k for which the equation has two distinct real roots”. The method is to set the discriminant greater than zero.
最常见的问法是“求k的取值范围，使该方程有两个不同实根”。解题方法就是令判别式大于零。
x² + kx + 3 = 0 has two distinct real roots ⇒ k² – 12 > 0
Therefore k < -2√3 or k > 2√3. If the question says “real roots”, use ≥ 0; if it says “no real roots”, use < 0.
因此k < -2√3 或 k > 2√3。如果题目说“有实根”，用 ≥ 0；如果题目说“无实根”，用 < 0。
You should also know the completed square form y = a(x – h)² + k. From this you can read the turning point (h, k) and the line of symmetry x = h.
你还应掌握配方法的形式 y = a(x – h)² + k。从中可以直接读出顶点(h, k)和对称轴 x = h。
The discriminant is also used to test whether a straight line meets a curve. Substitute the line equation into the curve, then check whether the resulting quadratic has two roots, one root, or no roots.
判别式也常用于判断直线与曲线是否相交。把直线方程代入曲线方程，再检查所得二次方程有两个根、一个根还是没有根。
2. Functions, Inverse Functions and Transformations | 函数、反函数与图像变换
Functions questions usually test domain and range, inverse functions, composite functions, and graph transformations. The most common mistake is forgetting the domain before finding an inverse.
函数题通常考查定义域与值域、反函数、复合函数和图像变换。最常见的错误是在求反函数之前忘记考虑定义域。
For example, if f(x) = 2x + 3, then its inverse is found by rearranging x = 2y + 3 to give y = (x – 3)/2. Always check that the range of the inverse equals the domain of the original function.
例如，若 f(x) = 2x + 3，求反函数时先写成 x = 2y + 3，再整理得 y = (x – 3)/2。同时要检查反函数的值域等于原函数的定义域。
Graph transformations are commonly tested in the form “describe the transformation that maps y = f(x) onto y = 2f(x + 1)”. The answer should include both the type and the direction.
图像变换常以“描述由 y = f(x) 到 y = 2f(x + 1) 的变换”这样的形式考查。答案必须同时说明变换类型和方向。
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y = f(x) + a: translation by a units in the y direction.
y = f(x) + a：沿y方向平移a个单位。
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y = f(x + a): translation by -a units in the x direction.
y = f(x + a)：沿x方向平移-a个单位。
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y = -f(x): reflection in the x-axis.
y = -f(x)：关于x轴反射。
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y = f(-x): reflection in the y-axis.
y = f(-x)：关于y轴反射。
3. Coordinate Geometry and Circles | 坐标几何与圆
Coordinate geometry questions often require you to find midpoints, lengths, gradients, and equations of lines. You must remember that two lines are perpendicular when the product of their gradients is -1.
坐标几何题通常要求你求中点、长度、斜率和直线方程。你必须记住：两条直线垂直时，它们的斜率乘积为 -1。
Circle questions are very common. The standard equation is (x – a)² + (y – b)² = r², where (a, b) is the centre and r is the radius.
圆的相关题目非常常见。标准方程为 (x – a)² + (y – b)² = r²，其中(a, b)是圆心，r是半径。
A typical question asks you to find the equation of the tangent to a circle at a given point. The radius to that point is perpendicular to the tangent, so first find the gradient of the radius, then take the negative reciprocal.
典型题目要求你求圆在某个已知点处的切线方程。过该点的半径与切线垂直，因此先求半径的斜率，再取负倒数。
m_radius × m_tangent = -1
Using m_tangent and the point, substitute into y – y₁ = m(x – x₁).
利用切线斜率m和已知点，代入 y – y₁ = m(x – x₁) 即可。
4. Sequences and Series | 数列与级数
Arithmetic and geometric sequences are tested every year. For an arithmetic sequence, the nth term is a + (n – 1)d and the sum of n terms is Sₙ = n/2(2a + (n – 1)d).
等差和等比数列每年都会考查。等差数列的第n项是 a + (n – 1)d，前n项和是 Sₙ = n/2(2a + (n – 1)d)。
For a geometric sequence, the nth term is arⁿ⁻¹ and the sum of n terms is Sₙ = a(1 – rⁿ)/(1 – r). When |r| < 1, the sum to infinity is a/(1 - r).
等比数列的第n项是 arⁿ⁻¹，前n项和是 Sₙ = a(1 – rⁿ)/(1 – r)。当 |r| < 1 时，无穷项和是 a/(1 - r)。
Example: the sequence 3, 8, 13, 18, … gives uₙ = 5n – 2. Its sum is Sₙ = n/2(2 × 3 + (n – 1) × 5) = n/2(5n + 1).
例如：数列3, 8, 13, 18, … 的通项是 uₙ = 5n – 2。其前n项和为 Sₙ = n/2(2 × 3 + (n – 1) × 5) = n/2(5n + 1)。
Common marks are lost when n is incorrectly substituted. Always check the first term and the first sum with your formula.
常见的失分点是把n代错。一定要用第一项和第一项和来检查公式是否正确。
5. Polynomials and the Binomial Theorem | 多项式与二项式定理
Polynomial questions use the factor theorem and the remainder theorem. The factor theorem says that if f(a) = 0, then (x – a) is a factor of f(x).
多项式题目会用到因式定理和余数定理。因式定理指出：如果 f(a) = 0，那么 (x – a) 是 f(x) 的一个因式。
A classic question is to factorise a cubic like x³ – 6x² + 11x – 6. Testing x = 1 gives 0, so (x – 1) is a factor. Dividing gives x² – 5x + 6, which factors as (x – 2)(x – 3).
经典题目是分解三次多项式 x³ – 6x² + 11x – 6。试 x = 1 得0，因此 (x – 1) 是因式。做除法得到 x² – 5x + 6，再分解为 (x – 2)(x – 3)。
Binomial expansion questions often ask for the first four terms of (1 + 2x)⁵. Using nCr coefficients gives:
二项式展开题常要求写出 (1 + 2x)⁵ 的前四项。利用组合数系数可得：
(1 + 2x)⁵ = 1 + 10x + 40x² + 80x³ + 80x⁴ + 32x⁵
If the expression is (a + b)ⁿ, remember that the powers of a decrease and the powers of b increase.
若展开式是 (a + b)ⁿ，要注意a的幂次下降，b的幂次上升。
6. Differentiation and Applications | 微分及其应用
Differentiation is the most heavily tested skill in A-level pure mathematics. You must know the power rule, product rule, quotient rule, and chain rule.
微分是A-level纯数学中考查最频繁的技能。你必须掌握幂法则、乘积法则、商法则和链式法则。
For example, if y = x³ – 6x² + 9x + 1, then dy/dx = 3x² – 12x + 9. Solving dy/dx = 0 gives stationary points at x = 1 and x = 3.
例如，若 y = x³ – 6x² + 9x + 1，则 y’ = 3x² – 12x + 9。令 y’ = 0，可得到 x = 1 和 x = 3 处的驻点。
To classify them, use the second derivative. At x = 1, d²y/dx² = 6x – 12 = -6, so it is a maximum. At x = 3, d²y/dx² = 6, so it is a minimum.
要判断驻点类型，使用二阶导数。在 x = 1 处，y” = 6x – 12 = -6，所以是极大值点；在 x = 3 处，y” = 6，所以是极小值点。
Tangent questions are just as common. If you need the tangent at a point, first find the gradient using dy/dx, then use y – y₁ = m(x – x₁).
切线题同样常见。若要求某点处的切线，先用 y’ 求斜率，再用 y – y₁ = m(x – x₁)。
Optimisation problems are also common. Write the quantity to be maximised or minimised as a single-variable function, differentiate, set the derivative to zero, and confirm whether it is a maximum or minimum.
最优化问题也很常见。先把要最大化或最小化的量写成单变量函数，求导后令导数为零，再确认是极大值还是极小值。
7. Integration and Areas | 积分与面积
Integration questions require you to find antiderivatives, evaluate definite integrals, and find areas. The basic rule is ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ -1.
积分题目要求你求原函数、计算定积分和求面积。基本公式是 ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C，其中 n ≠ -1。
A typical area question gives a curve and a range of x values. For example, the area between x = 2 and x = 3 under y = x² – 2x is:
典型的面积题会给出曲线和x的范围。例如，曲线 y = x² – 2x 在 x = 2 到 x = 3 之间的面积为：
∫₂³ (x² – 2x) dx = [x³/3 – x²]₂³ = 4/3
If the curve goes below the x-axis, the definite integral gives a negative value. To find the actual area in that interval, take the absolute value or integrate the negative of the function.
如果曲线在x轴下方，定积分会得到负值。要求该区间内的实际面积，需要取绝对值，或者对被积函数取相反数后再积分。
When finding the area between two curves, integrate (upper curve – lower curve) between their points of intersection.
当求两条曲线之间的面积时，在两条曲线的交点之间积分（上方曲线 – 下方曲线）。
8. Trigonometry | 三角函数
Trigonometry questions typically test identities, equations, and transformations. The most important identity is sin²θ + cos²θ = 1.
三角函数题通常考查恒等式、方程和图像变换。最重要的恒等式是 sin²θ + cos²θ = 1。
You must also know tan θ = sin θ / cos θ, and the exact values of sin, cos and tan at 0°, 30°, 45°, 60° and 90°.
你还需要知道 tan θ = sin θ / cos θ，以及 0°、30°、45°、60°、90° 时sin、cos、tan的准确值。
A common equation question is: solve 2cos²θ – cosθ – 1 = 0 for 0 ≤ θ ≤ 360°. Factorising gives (2cosθ + 1)(cosθ – 1) = 0.
常见的方程题是：在0° ≤ θ ≤ 360°内解 2cos²θ – cosθ – 1 = 0。因式分解得 (2cosθ + 1)(cosθ – 1) = 0。
So cosθ = -1/2 or cosθ = 1. The solutions are θ = 120°, 240° and θ = 0°, 360°. Always check whether the interval includes the endpoints.
因此 cosθ = -1/2 或 cosθ = 1。解为 θ = 120°、240° 以及 θ = 0°、360°。一定要检查区间是否包含端点。
Compound angle formulas are also common. Make sure you can use sin(A ± B), cos(A ± B) and tan(A ± B) without mixing signs.
复角公式也很常见。务必能够正确使用 sin(A ± B)、cos(A ± B) 和 tan(A ± B)，不要搞错符号。
9. Exponentials and Logarithms | 指数函数与对数
Exponential and logarithmic questions test the laws of logs and the relationship between exponentials and logs. The key idea is that logₐx is the inverse of aˣ.
指数与对数题考查对数运算法则，以及指数与对数的互逆关系。核心思想是 logₐx 是 aˣ 的反函数。
Common log laws are logₐ(mn) = logₐm + logₐn, logₐ(m/n) = logₐm – logₐn, and logₐmⁿ = n log
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