📚 Year 1 Edexcel Further Maths: Core Pure Year 1 | Edexcel 一年级进阶数学：核心纯数第一年
Edexcel Further Mathematics Year 1 builds on A Level Mathematics by introducing more abstract tools such as complex numbers, matrices, proof by induction, and vectors in three dimensions. This article summarises the Core Pure Year 1 topics that appear most often in exam questions, with key formulas and revision points.
Edexcel 进阶数学第一年在普通 A Level 数学基础上引入复数、矩阵、归纳法证明以及三维向量等更抽象的工具。本文总结核心纯数第一年考试中最常出现的主题、关键公式和复习要点。
1. Complex Numbers | 复数
n be written in the form z = a + bi, where a is the real part and b is the imaginary part.
复数通过定义虚数单位 i（满足 i² = -1）扩展了实数系。每个复数都可以写成 z = a + bi 的形式，其中 a 是实部，b 是虚部。
z = a + bi, z* = a − bi, |z| = √a² + b²
nother. Multiplying a complex number by its conjugate gives a real number: z z* = a² + b².
你必须会进行复数的加、减、乘运算，并利用共轭复数 z* = a − bi 进行除法。一个复数与其共轭相乘得到实数：z z* = a² + b²。
- Solve quadratic equations with complex roots, e.g. x² + 4x + 13 = 0 → x = −2 ± 3i | 解具有复数根的二次方程，例如 x² + 4x + 13 = 0 → x = −2 ± 3i
- State the real and imaginary parts and simplify powers of i using i² = −1 | 写出实部和虚部，并利用 i² = −1 化简 i 的幂
2. Argand Diagrams | 阿尔冈图
nt or vector in a plane with horizontal real axis and vertical imaginary axis. The modulus |z| , and the argument arg(z) is the angle measured anticlockwise from the positive real axis.
阿尔冈图把复数 z = a + bi 表示为平面上的点或向量，横轴是实轴，纵轴是虚轴。模 |z| 是到原点的距离，辐角 arg(z) 是从正实轴逆时针测量的角度。
|z| = √a² + b², arg(z) = θ, tan θ = b/a
s multiplication and division easier: multiply the moduli and add the arguments when multiplying, or divide the moduli and subtract the arguments when dividing.
把复数写成模-辐角形式 z = r(cos θ + i sin θ) 会使乘除运算更简单：乘法时模相乘、辐角相加；除法时模相除、辐角相减。
- Plot complex numbers and their conjugates as reflections in the real axis | 在阿尔冈图上绘制复数及其共轭，共轭是关于实轴的反射
- Use the principal argument, usually −π < θ ≤ π | 使用主辐角，通常 −π < θ ≤ π
3. Roots of Polynomials | 多项式根
with roots α and β, the sum and product of roots are related to the coefficients. These relationships allow you to find unknown coefficients or to form a new equation with transformed roots.
对于具有根 α 和 β 的二次方程 ax² + bx + c = 0，根的和与积与系数之间有关系。这些关系可以帮助你求未知系数，或构造具有变换根的新方程。
α + β = −b/a, αβ = c/a
with roots α, β and γ, you need all three symmetric sums. These are often tested alongside finding a cubic from given roots.
对于具有根 α、β 和 γ 的三次方程 ax³ + bx² + cx + d = 0，你需要掌握全部三个对称和。这些内容常与由已知根求三次方程一起考查。
α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a
- Use root relations to find expressions such as α² + β² = (α + β)² − 2αβ | 利用根的关系求表达式，例如 α² + β² = (α + β)² − 2αβ
- Form new equations whose roots are raised by a constant, squared, or reciprocated | 构造新方程，其根是原根加常数、平方或取倒数
4. Matrices | 矩阵
tion, subtraction, scalar multiplication, and matrix multiplication. Matrix multiplication is not commutative, so AB is not generally equal to BA.
矩阵是数字的矩形阵列。在核心纯数第一年，你会学习 2×2 和 3×3 矩阵的加法、减法、数乘和矩阵乘法。矩阵乘法不满足交换律，因此 AB 通常不等于 BA。
(AB)ᵢⱼ = Σ Aᵢₖ Bₖⱼ
erminant is zero, the matrix is singular and has no inverse. Otherwise, the inverse is found by swapping the diagonal entries, changing the signs of the off-diagonal entries, and dividing by the determinant.
对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det M = ad − bc。如果行列式为零，矩阵是奇异矩阵，没有逆矩阵。否则，求逆矩阵的方法是交换主对角线元素、改变副对角线元素符号，再除以行列式。
M⁻¹ = (1/det M) × [[d, −b], [−c, a]]
- Check that MM⁻¹ = I to verify an inverse | 通过 MM⁻¹ = I 验证逆矩阵
- Understand that det(MN) = det M × det N | 理解 det(MN) = det M × det N
5. Linear Transformations | 线性变换
ansformation in the plane. The image of the point (x, y) is obtained by multiplying the matrix by the column vector [[x], [y]]. The columns of the transformation matrix are the images of the basis vectors (1, 0) and (0, 1).
2×2 矩阵可以表示平面中的线性变换。点 (x, y) 的像通过矩阵乘以列向量 [[x], [y]] 得到。变换矩阵的两列分别是基向量 (1, 0) 和 (0, 1) 的像。
[[a, b], [c, d]] [[x], [y]] = [[ax + by], [cx + dy]]
Common linear transformations include rotations about the origin, reflections in lines through the origin, enlargements centred at the origin, and shears. You should be able to identify a transformation from its matrix and find the matrix for a given transformation.
常见的线性变换包括绕原点的旋转、关于过原点直线的反射、以原点为中心的放大以及剪切变换。你应当能够从矩阵识别变换，以及根据给定变换求矩阵。
- Rotation by θ anticlockwise: [[cos θ, −sin θ], [sin θ, cos θ]] | 逆时针旋转 θ：[[cos θ, −sin θ], [sin θ, cos θ]]
- t points satisfy M x = x; invariant lines may have direction vectors unchanged up to scale | 不变点满足 M x = x；不变线的方向向量在变换后可能只改变比例
6. Series | 级数
In Core Pure Year 1 you use standard results for the sums of positive integers, squares, and cubes to evaluate finite series. These standard sums are often combined with algebraic manipulation before applying the formulas.
在核心纯数第一年，你会使用正整数、平方和立方的标准求和结果来计算有限级数。这些标准求和式通常在应用公式前需要先进行代数变形。
∑r = n(n+1)/2, ∑r² = n(n+1)(2n+1)/6, ∑r³ = [n(n+1)/2]²
ts and then applying the standard results. Always write each term separately and factorise your final answer where possible.
题目可能要求求 ∑(r+2)(r−1) 之类的和，方法是先展开括号，再应用标准结果。始终把每一项分开写出，并在可能时对最终答案进行因式分解。
- Remember the lower limit may not be 1; use ∑ from r=1 to n minus ∑ from r=1 to k−1 | 注意下限可能不是 1；用从 r=1 到 n 的和减去从 r=1 到 k−1 的和
- Look for hidden standard sums in expressions such as ∑(2r + 3) | 在 ∑(2r + 3) 等表达式中识别隐藏的标准求和式
7. Proof by Induction | 归纳法证明
ctive step you assume the result is true for n = k and show it must be true for n = k + 1.
归纳法证明用于证明对所有正整数 n 都成立的命题。它分为四部分：基础情形、归纳假设、归纳步骤和结论。在归纳步骤中，假设结果对 n = k 成立，并证明它一定对 n = k + 1 也成立。
If f(k+1) − f(k) is divisible by d, then f(k+1) is divisible by d when f(k) is.
Common induction questions include proving summation formulas, divisibility results, matrix powers, and recurrence relations. Always write a clear conclusion such as 'P(k+1) follows from P(k), so by induction P(n) is true for all positive integers n.'
常见的归纳法题目包括证明求和公式、整除性、矩阵的幂以及递推关系。始终写出清晰的结论，例如“由 P(k) 推出 P(k+1)，因此根据归纳法 P(n) 对所有正整数 n 都成立”。
- Basis case: verify n = 1 | 基础情形：验证 n = 1
- Inductive hypothesis: assume true for n = k | 归纳假设：假设 n = k 时命题成立
- Inductive step: prove for n = k + 1 using the assumption | 归纳步骤：利用假设证明 n = k + 1 时成立
8. Vectors | 向量
In Further Maths Year 1, vectors are extended to three dimensions. You work with position vectors, magnitude, unit vectors, and the scalar (dot) product. The scalar product is used to find the angle between two vectors and to determine perpendicularity.
在进阶数学第一年中，向量扩展到三维。你会学习位置向量、模、单位向量以及标量积（点积）。标量积用于求两个向量之间的夹角以及判断垂直关系。
a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
d b is a direction vector. You may need to find the angle between two lines or determine whether two lines intersect.
三维中直线的向量方程是 r = a + λ b，其中 a 是直线上一点，b 是方向向量。你可能需要求两条直线的夹角或判断两条直线是否相交。
r = a + λ b
- Two vectors are perpendicular if a · b = 0 | 如果 a · b = 0，则两个向量垂直
- Unit vector in direction a: â = a / |a| | 沿 a 方向的单位向量：â = a / |a|
9. First Order Differential Equations | 一阶微分方程
= f(x)g(y), then you can rearrange so that all y terms are on one side and all x terms on
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