📚 Year 12 SQA Advanced Higher Mathematics: Formula & Theorem Quick Reference Guide | SQA 进阶数学公式定理速查手册
This quick reference guide collates the essential formulas, theorems and standard results required for the SQA Advanced Higher Mathematics course. Use it to review key concepts before your assessments and examinations.
本速查手册汇集了 SQA 进阶数学（Advanced Higher）课程所需的核心公式、定理与标准结论。可用于考试前快速回顾关键要点。
1. Algebra and Partial Fractions | 代数与部分分式
For any quadratic equation ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives a repeated real root, and Δ < 0 gives complex conjugate roots.
对于一元二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有二重实根，Δ < 0 有一对共轭复根。
Partial fractions expansions are built from the denominator’s factors. For distinct linear factors: 1 / [(px+q)(rx+s)] ≡ A/(px+q) + B/(rx+s). For a repeated linear factor (px+q)ⁿ, include terms A₁/(px+q) + A₂/(px+q)² + … + Aₙ/(px+q)ⁿ. For an irreducible quadratic factor (ax²+bx+c), the term takes the form (Bx+C)/(ax²+bx+c).
部分分式展开基于分母的因式。对于不同的线性因式：1 / [(px+q)(rx+s)] ≡ A/(px+q) + B/(rx+s)。对于重复线性因式 (px+q)ⁿ，依次包含 A₁/(px+q) + A₂/(px+q)² + … + Aₙ/(px+q)ⁿ。对于不可约二次因式 (ax²+bx+c)，项的形式为 (Bx+C)/(ax²+bx+c)。
2. Binomial Theorem and Series | 二项式定理与级数
The general binomial expansion for a rational index n is valid when |x| < 1: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …. For (a + b)ⁿ with positive integer n, the expansion is Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ.
有理数指数 n 的一般二项式展开在 |x| < 1 时成立：(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …。对于正整数指数 n，有 (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ。
Frequently used series: 1/(1−x) = 1 + x + x² + x³ + … for |x| < 1, and 1/(1+x)² = 1 − 2x + 3x² − 4x³ + ….
常用级数：|x| < 1 时 1/(1−x) = 1 + x + x² + x³ + …；1/(1+x)² = 1 − 2x + 3x² − 4x³ + …。
3. Complex Numbers and De Moivre | 复数与棣莫弗定理
A complex number in Cartesian form is z = a + bi. Its modulus is r = |z| = √(a² + b²) and its argument θ satisfies tan θ = b/a. The polar form is z = r (cos θ + i sin θ), often written as z = r e^(iθ).
复数的笛卡儿形式为 z = a + bi。其模为 r = |z| = √(a² + b²)，辐角 θ 满足 tan θ = b/a。极坐标为 z = r (cos θ + i sin θ)，常记作 z = r e^(iθ)。
Multiplication and division in polar form: z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)] and z₁/z₂ = (r₁/r₂) [cos(θ₁−θ₂) + i sin(θ₁−θ₂)]. De Moivre’s theorem states: zⁿ = rⁿ (cos nθ + i sin nθ) for any integer n.
极坐标形式的乘除法：z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)]；z₁/z₂ = (r₁/r₂) [cos(θ₁−θ₂) + i sin(θ₁−θ₂)]。棣莫弗定理：对于任意整数 n，zⁿ = rⁿ (cos nθ + i sin nθ)。
The n-th roots of z = r (cos θ + i sin θ) are given by zₖ^(1/n) = r^(1/n) [cos((θ+2πk)/n) + i sin((θ+2πk)/n)] for k = 0, 1, 2, …, n−1.
复数 z 的 n 次方根为 zₖ^(1/n) = r^(1/n) [cos((θ+2πk)/n) + i sin((θ+2πk)/n)]，其中 k = 0, 1, 2, …, n−1。
4. Vectors: Dot, Cross Product, Lines and Planes | 向量：点积、叉积、直线与平面
For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the dot product is a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. The cross product is a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), which is perpendicular to both a and b.
对于向量 a = (a₁, a₂, a₃) 与 b = (b₁, b₂, b₃)，点积为 a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ。叉积为 a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)，其结果垂直于 a 和 b。
The vector equation of a line passing through point A with direction vector v is r = a + t v, t ∈ ℝ. The Cartesian form is (x−x₀)/v₁ = (y−y₀)/v₂ = (z−z₀)/v₃.
过定点 A 且方向向量为 v 的直线向量方程为 r = a + t v，t ∈ ℝ。笛卡儿形式为 (x−x₀)/v₁ = (y−y₀)/v₂ = (z−z₀)/v₃。
A plane with normal vector n = (A, B, C) passing through point (x₀, y₀, z₀) has equation A(x−x₀) + B(y−y₀) + C(z−z₀) = 0, or r · n = a · n. The distance from a point to a plane can be found by projection.
以 n = (A, B, C) 为法向量且过点 (x₀, y₀, z₀) 的平面方程为 A(x−x₀) + B(y−y₀) + C(z−z₀) = 0，也可写为 r · n = a · n。点到平面的距离可由投影求得。
5. Matrices and Systems of Equations | 矩阵与方程组
For a 2×2 matrix A = [a b; c d], the determinant is det A = ad − bc. The inverse, provided det A ≠ 0, is A⁻¹ = 1/(ad−bc) [d −b; −c a]. A 3×3 determinant can be expanded by any row or column using cofactors.
对于 2×2 矩阵 A = [a b; c d]，行列式为 det A = ad − bc。若 det A ≠ 0，其逆矩阵为 A⁻¹ = 1/(ad−bc) [d −b; −c a]。3×3 行列式可通过某行或某列的代数余子式展开计算。
A system of linear equations can be written as A x = b. Gaussian elimination reduces the augmented matrix to row-echelon form to solve for the variables. The system has a unique solution if and only if det A ≠ 0.
线性方程组可写成矩阵形式 A x = b。高斯消元法将增广矩阵化为行阶梯形以求解未知数。当且仅当 det A ≠ 0 时方程组有唯一解。
Transformations in the plane are represented by 2×2 matrices: rotation by angle θ is [cosθ −sinθ; sinθ cosθ], reflection in the line y = (tanθ)x uses a similar form, and scaling/dilations are diagonal matrices.
平面上的变换可用 2×2 矩阵表示：旋转 θ 角的矩阵为 [cosθ −sinθ; sinθ cosθ]，关于直线 y = (tanθ)x 的反射也有相似形式，缩放/拉伸对应对角矩阵。
6. Differentiation Techniques | 微分技巧
Chain rule: dy/dx = dy/du × du/dx. Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v − uv’) / v². For implicit differentiation, differentiate both sides with respect to x and apply the chain rule to terms involving y.
链式法则：dy/dx = dy/du × du/dx。乘积法则：d/dx (uv) = u’v + uv’。商法则：d/dx (u/v) = (u’v − uv’) / v²。隐函数微分则是对等式两边关于 x 求导，并对含有 y 的项应用链式法则。
Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt). Logarithmic differentiation takes ln of both sides and is useful for expressions with variable exponents.
参数方程求导：若 x = f(t), y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。对数微分法将两边取自然对数，适用于变量作指数的情形。
Standard derivatives:
基本导数公式：
d/dx (xⁿ) = n xⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (aˣ) = aˣ ln a, d/dx (ln x) = 1/x
d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x
d/dx (arcsin x) = 1/√(1−x²), d/dx (arctan x) = 1/(1+x²)
d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x
7. Integration Methods | 积分方法
Integration is the reverse of differentiation. Standard integrals:
积分是微分的逆运算。基本积分公式：
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ / ln a + C
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C
∫ 1/√(a²−x²) dx = arcsin(x/a) + C, ∫ 1/(a²+x²) dx = (1/a) arctan(x/a) + C
Substitution: use u = g(x) so that du = g'(x) dx. For definite integrals, change the limits accordingly.
换元积分法：令 u = g(x)，则 du = g'(x) dx。对于定积分，需要相应改变积分限。
Integration by parts: ∫ u dv = uv − ∫ v du. A useful strategy is LIATE (Log, Inverse trig, Algebraic, Trig, Exponential) to choose u. Repeated integration by parts may be needed.
分部积分法：∫ u dv = uv − ∫ v du。可参考 LIATE 顺序（对数、反三角、代数、三角、指数）选择 u。有时需多次使用分部积分。
Partial fractions integration: express a rational function as a sum of simpler fractions and integrate term by term. Trigonometric integrals often use identities like sin²x = ½(1−cos 2x).
部分分式积分：将有理函数拆分为简单分式之和，然后逐项积分。三角积分常用恒等式，如 sin²x = ½(1−cos 2x)。
8. Differential Equations | 微分方程
A first-order separable equation has the form dy/dx = g(x)h(y). Separate variables: ∫ 1/h(y) dy = ∫ g(x) dx. A linear first-order equation dy/dx + P(x)y = Q(x) can be solved using an integrating factor μ(x) = e^(∫ P dx).
一阶可分离变量的微分方程形如 dy/dx = g(x)h(y)。分离变量得 ∫ 1/h(y) dy = ∫ g(x) dx。一阶线性方程 dy/dx + P(x)y = Q(x) 可使用积分因子 μ(x) = e^(∫ P dx) 求解。
Second-order linear homogeneous equations with constant coefficients: a d²y/dx² + b dy/dx + c y = 0. The auxiliary equation is a m² + b m + c = 0. For real distinct roots m₁, m₂, the general solution is y = A e^(m₁x) + B e^(m₂x). For repeated root m, use y = (A + Bx) e^(m x). For complex roots α ± iβ, the solution is y = e^(αx) (A cos βx + B sin βx).
二阶常系数线性齐次方程：a d²y/dx² + b dy/dx + c y = 0。辅助方程为 a m² + b m + c = 0。若有两个不等实根 m₁, m₂，通解为 y = A e^(m₁x) + B e^(m₂x)；重根 m 时，解为 y = (A + Bx) e^(m x)；复根 α ± iβ 时，解为 y = e^(αx) (A cos βx + B sin βx)。
Non-homogeneous second-order ODEs: find a particular integral yₚ based on the form of f(x). Common choices: for polynomial, try polynomial of the same degree; for e^(kx), try C e^(kx); for sin/cos, try C cos ωx + D sin ωx. Then general solution is y = y_c (complementary) + yₚ.
二阶非齐次常微分方程：根据 f(x) 的形式设定特解 yₚ。常见情形：多项式则试设同次多项式；e^(kx) 则试设 C e^(kx)；正弦/余弦则试设 C cos ωx + D sin ωx。通解为 y = y_c（齐次通解）+ yₚ。
9. Sequences, Summation and Maclaurin Series | 序列、求和与麦克劳林级数
Arithmetic sequence: n-th term uₙ = a + (n−1)d, sum to n terms Sₙ = n/2 [2a + (n−1)d]. Geometric sequence: n-th term uₙ = a rⁿ⁻¹, sum to n terms Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1, and sum to infinity S∞ = a/(1−r) when |r| < 1.
等差数列：通项 uₙ = a + (n−1)d，前 n 项和 Sₙ = n/2 [2a + (n−1)d]。等比数列：通项 uₙ = a rⁿ⁻¹，前 n 项和（r ≠ 1）Sₙ = a(1−rⁿ)/(1−r)，当 |r| < 1 时无穷和为 S∞ = a/(1−r)。
Standard summation formulas: Σᵢ₌₁ⁿ i = n(n+1)/2, Σ i² = n(n+1)(2n+1)/6, Σ i³ = [n(n+1)/2]².
常见求和公式：Σᵢ₌₁ⁿ i = n(n+1)/2，Σ i² = n(n+1)(2n+1)/6，Σ i³ = [n(n+1)/2]²。
The Maclaurin series of a function f(x) is its Taylor expansion about x = 0: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …. Key expansions include:
函数 f(x) 的麦克劳林级数是其在 x = 0 处的泰勒展开：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。关键展开式有：
eˣ = 1 + x + x²/2! + x³/3! + … valid for all x
sin x = x − x³/3! + x⁵/5! − … cos x = 1 − x²/2! + x⁴/4! − …
ln(1+x) = x − x²/2 + x³/3 − … for −1 < x ≤ 1
10. Proof by Induction | 数学归纳法
Proof by mathematical induction verifies that a statement P(n) holds for all positive integers n. It follows three steps: (1) Base case – show P(1) is true; (
Published by TutorHao | Year 12 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导