ESAT Mathematics 1: Core Pure Maths Topics Explained | ESAT 数学一：纯数学核心考点全解析

📚 ESAT Mathematics 1: Core Pure Maths Topics Explained | ESAT 数学一：纯数学核心考点全解析

The ESAT Mathematics 1 paper tests the core pure mathematics skills expected of strong A-level students. This article breaks down every major topic, showing the essential results and common traps you must avoid.

ESAT 数学一试卷考察优秀A-Level学生应具备的纯数学核心技能。本文将拆解每一个主要考点，呈现必备结论与常见陷阱。

1. Polynomials and Quadratic Functions | 多项式与二次函数

A quadratic in the form ax² + bx + c = 0 is solved by factoring, completing the square, or using the formula. The value of the discriminant determines the nature of the roots.

对于 ax² + bx + c = 0 形式的二次方程，可通过因式分解、配方法或求根公式求解。判别式的值决定根的性质。

Δ = b² − 4ac

If Δ > 0, the equation has two distinct real roots.

若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is one repeated real root.

若 Δ = 0，方程有一个二重实数根。
If Δ < 0, there are no real roots, but two complex conjugate roots.

若 Δ < 0，方程无实数根，但有两个共轭复数根。

The sum and product of roots for α and β are useful: α + β = −b/a and αβ = c/a. For higher-degree polynomials, use factor theorems and polynomial division to find factors.

对于根 α 与 β，根与系数关系非常实用：α + β = −b/a，αβ = c/a。对于高次多项式，可使用因式定理与多项式除法寻找因式。

2. Inequalities and Sets | 不等式与解集

Linear inequalities are solved exactly like equations, except that multiplying or dividing by a negative number reverses the inequality sign. Quadratic inequalities require sketching the parabola or testing intervals between critical points.

线性不等式的求解与方程类似，但两边乘除负数时必须改变不等号方向。二次不等式需要画抛物线草图或在临界点之间进行区间测试。

(x − a)(x − b) < 0 ⇒ a < x < b (for a < b)

For rational inequalities, move all terms to one side, combine into a single fraction, find critical points from numerator and denominator, then test intervals.

对于分式不等式，将所有项移到同一边，合并为一个分式，由分子分母找出临界点，再对各区间进行测试。

Common set notation includes ℝ for real numbers, ℤ for integers, and interval notation such as [−2, 5). Always write solutions in the requested form.

常见集合符号包括实数集 ℝ、整数集 ℤ，以及区间记号如 [−2, 5)。务必按题目要求的形式写出解集。

3. Coordinate Geometry and Lines | 坐标几何与直线

The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²], and the midpoint is the average of the coordinates.

两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]，中点坐标为两点坐标的平均值。

gradient m = (y₂ − y₁) / (x₂ − x₁)

The equation of a line can be written as y − y₁ = m(x − x₁), or ax + by + c = 0. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

直线方程可写为点斜式 y − y₁ = m(x − x₁)，也可写为一般式 ax + by + c = 0。平行直线斜率相等；垂直直线满足 m₁ × m₂ = −1。

The distance from a point (x₁, y₁) to a line ax + by + c = 0 is |ax₁ + by₁ + c| / √(a² + b²). Memorise this formula to save time in the exam.

点 (x₁, y₁) 到直线 ax + by + c = 0 的距离为 |ax₁ + by₁ + c| / √(a² + b²)。熟记此公式可在考试中节省时间。

4. Exponential and Logarithmic Functions | 指数与对数函数

The exponential function eˣ is special because its derivative is itself. Natural logarithms are the inverse of eˣ, so ln(eˣ) = x and e^(ln x) = x.

指数函数 eˣ 的特殊之处在于其导数仍是自身。自然对数是 eˣ 的反函数，因此 ln(eˣ) = x，且 e^(ln x) = x。

log(ab) = log a + log b

log(ab) = log a + log b
log(a/b) = log a − log b

log(a/b) = log a − log b
log(aⁿ) = n log a

log(aⁿ) = n log a

Remember the change of base rule: log_a(b) = ln b / ln a. When solving equations, always check that arguments of logs are positive.

牢记换底公式：log_a(b) = ln b / ln a。解方程时，务必检验对数的真数均为正数。

5. Functions and Transformations | 函数与图像变换

A function maps each input to exactly one output. Its domain is the set of allowed inputs, and its range is the set of possible outputs. A function has an inverse only if it is one-to-one.

函数将每个输入映射到唯一输出。定义域是允许的输入集合，值域是可能输出的集合。只有一一映射的函数才有反函数。

Transformations follow a clear pattern:

图像变换遵循清晰的模式：

y = f(x) + a: shift up by a.

y = f(x) + a: 向上平移 a 个单位。
y = f(x − a): shift right by a.

y = f(x − a): 向右平移 a 个单位。
y = −f(x): reflect in the x-axis.

y = −f(x): 关于 x 轴反射。
y = f(−x): reflect in the y-axis.

y = f(−x): 关于 y 轴反射。
y = k f(x): vertical stretch by factor k.

y = k f(x): 纵向拉伸 k 倍。

Pay attention to order: horizontal shifts and stretches can be tricky when combined. Test a few key points to verify your sketch.

注意变换顺序：水平平移与伸缩组合时容易出错。先取几个关键点验证草图即可。

6. Trigonometry | 三角函数

Exact values for common angles must be memorised: sin 30° = ½, cos 60° = ½, tan 45° = 1. The Pythagorean identity sin²θ + cos²θ = 1 is used in many simplifications.

常见角的精确值必须熟记：sin 30° = ½，cos 60° = ½，tan 45° = 1。平方关系 sin²θ + cos²θ = 1 常用于化简。

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

For equations such as a sin θ + b cos θ = c, use the harmonic form R sin(θ + α) where R = √(a² + b²) and tan α = b/a (with correct quadrant). Always find all solutions within the given interval.

对于形如 a sin θ + b cos θ = c 的方程，可使用辅助角公式 R sin(θ + α)，其中 R = √(a² + b²)，tan α = b/a（注意象限）。务必在给定区间内求出所有解。

7. Differentiation | 微分

Differentiation measures rates of change. The derivative of xⁿ is nxⁿ⁻¹. The derivative of a sum is the sum of the derivatives, and constants factor out.

微分用于衡量变化率。xⁿ 的导数为 nxⁿ⁻¹。多项式和的导数等于各项导数的和，常数因子可提出。

d/dx (sin x) = cos x and d/dx (cos x) = −sin x

d/dx (sin x) = cos x，d/dx (cos x) = −sin x
d/dx (eˣ) = eˣ and d/dx (ln x) = 1/x

d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x

The chain rule states dy/dx = dy/du × du/dx. Use it for composite functions, such as differentiating (x² + 1)⁵.

链式法则为 dy/dx = dy/du × du/dx。求复合函数如 (x² + 1)⁵ 的导数时务必使用。

Stationary points occur where dy/dx = 0. Classify them using the second derivative: positive gives a local minimum, negative gives a local maximum, zero suggests a point of inflection.

驻点出现在 dy/dx = 0 处。可通过二阶导数分类：正值为极小值点，负值为极大值点，零可能为拐点。

8. Integration | 积分

Integration is the reverse of differentiation. For xⁿ, when n ≠ −1, the integral is xⁿ⁺¹/(n+1) + C. The constant C is essential for indefinite integrals.

积分是微分的逆运算。对于 xⁿ，当 n ≠ −1 时，其不定积分为 xⁿ⁺¹/(n+1) + C。不定积分必须包含常数 C。

∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C

∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C

To integrate composite functions, use integration by substitution or recognition. Definite integrals calculate the area between a curve and the x-axis. If the curve crosses the axis, split the integral and add absolute values.

求复合函数的积分时，可使用换元积分法或直接识别法。定积分可计算曲线与 x 轴之间的面积。若曲线穿过 x 轴，需分段积分并取绝对值相加。

The area between two curves is ∫ (y_top − y_bottom) dx over the appropriate interval. Always identify intersection points first.

两曲线之间的面积为 ∫ (y_上 − y_下) dx 在对应区间上的积分。务必先求出交点坐标。

9. Sequences and Series | 数列与级数

Arithmetic sequences have a constant difference d. The n-th term is a + (n−1)d and the sum of the first n terms is n/2[2a + (n−1)d].

等差数列具有公差 d。其第 n 项为 a + (n−1)d，前 n 项和为 n/2[2a + (n−1)d]。

Geometric sequences have a constant ratio r. The n-th term is arⁿ⁻¹. The sum of the first n terms is a(1 − rⁿ)/(1 − r) for r ≠ 1. An infinite geometric series converges to a/(1 − r) when |r| < 1.

等比数列具有公比 r。其第 n 项为 arⁿ⁻¹。当 r ≠ 1 时，前 n 项和为 a(1 − rⁿ)/(1 − r)。当 |r| < 1 时，无穷等比级数收敛于 a/(1 − r)。

Binomial expansion states:

二项式展开公式为：

(a + b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ

where C(n,r) = n! / [r!(n−r)!]. For small values of x and rational n, use the infinite binomial series with the appropriate range of validity.

其中 C(n,r) = n! / [r!(n−r)!]。当 x 很小且 n 为有理数时，可使用带有适用范围的无穷二项式级数。

10. Vectors in Pure Maths | 向量基础

A vector has both magnitude and direction. In component form, a = x i + y j + z k. The magnitude is |a| = √(x² + y² + z²).

向量同时具有大小和方向。在分量形式中，a = x i + y j + z k。模长为 |a| = √(x² + y² + z²)。

To add vectors, add components. The scalar product is a · b = |a||b| cos θ, which gives the angle between two vectors. Two vectors are perpendicular when a · b = 0.

向量加法即对应分量相加。数量积为 a · b = |a||b| cos θ，可用于求两向量夹角。当 a · b = 0 时两向量垂直。

For position vectors, the vector from A to B is OB − OA. If points lie on a straight line, their position vectors satisfy a linear relationship with constant coefficients summing to 1.

对于位置向量，由 A 到 B 的向量为 OB − OA。若点共线，则位置向量满足系数和为 1 的线性关系。

11. Curve Sketching and Graph Analysis | 曲线绘制与图像分析

Sketching a graph requires domain, intercepts, asymptotes, stationary points, and behaviour as x → ±∞. Always label axes and key coordinates.

绘制函数图像需要考察定义域、截距、渐近线、驻点以及当 x → ±∞ 时的趋势。务必标注坐标轴与关键坐标点。

Vertical asymptotes occur where the denominator is zero. Horizontal asymptotes are found by taking limits as x → ±∞. Oblique asymptotes occur when the degree of the numerator is one more than the denominator, found by polynomial division.

垂直渐近线出现在分母为零处。水平渐近线通过令 x → ±∞ 求极限得到。当分子次数比分母高一次时存在斜渐近线，可通过多项式除法求得。

Sign diagrams help determine where the function is positive or negative. Combine this with derivative information to produce an accurate sketch quickly.

符号图有助于判断函数在哪些区间为正或为负。将符号图与导数信息结合，可快速绘制准确草图。

12. Numerical Methods | 数值方法

Numerical methods are used when equations cannot be solved exactly. The sign change method finds a root if f(a) and f(b) have opposite signs and f is continuous on [a, b].

当方程无法精确求解时使用数值方法。若 f 在 [a, b] 上连续且 f(a) 与 f(b) 异号，则可用变号法求根。

Newton-Raphson iteration is:

牛顿-拉弗森迭代公式为：

xₙ₊₁ = xₙ − f(xₙ) / f′(xₙ)

Choose a starting value x₀ close to the root, then iterate. The trapezium rule approximates the area under a curve using strips of equal width:

选取靠近根的初始值 x₀，然后迭代。梯形法用等宽条带近似计算曲线下的面积：

∫ₐᵇ f(x) dx ≈ h/2 [y₀ + yₙ + 2(y₁ + y₂ + ⋯ + yₙ₋₁)]

where h = (b − a)/n. Larger n gives a more accurate approximation. Understand the error direction: for a concave curve the trapezium rule underestimates.

其中 h = (b − a)/n。n 越大近似越精确。还需理解误差方向：对于上凸曲线，梯形法会低估面积。

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ESAT Mathematics 1: Core Pure Maths Topics Explained | ESAT 数学一：纯数学核心考点全解析