📚 Year 10 SQA Advanced Mathematics: Formula & Theorem Quick Reference Handbook | Year 10 SQA 进阶数学：公式定理速查手册
Welcome to your Year 10 SQA Advanced Mathematics quick reference guide. This handbook brings together the essential formulas, theorems and key concepts across algebra, trigonometry, geometry, vectors, statistics, and sequences – everything you need for rapid revision and confident problem‑solving. Use it alongside your class notes, and practise applying each formula until it becomes second nature.
欢迎使用您的 Year 10 SQA 进阶数学速查手册。本手册汇集了代数、三角学、几何、向量、统计学与数列中最核心的公式、定理和关键概念，是您快速复习和自信解题的得力助手。请结合课堂笔记使用，并反复练习应用每条公式，直至熟练于心。
1. Algebraic Fractions & Operations | 代数分式与运算
Factorising is the reverse of expanding. The difference of two squares follows a² – b² = (a – b)(a + b), while a simple trinomial can often be written as (x + p)(x + q) where p+q equals the coefficient of x and pq equals the constant term.
因式分解是展开的逆运算。平方差公式为 a² – b² = (a – b)(a + b)，简单的三项式常可写成 (x + p)(x + q)，其中 p+q 等于 x 的系数，pq 等于常数项。
To simplify algebraic fractions, first factorise the numerator and denominator completely, then cancel common factors. For addition or subtraction, find a common denominator – often the product of the individual denominators.
化简代数分式时，应先将分子与分母彻底因式分解，然后约去公因式。做加减法时，先找到公分母，通常是各分母的乘积。
(a/b) + (c/d) = (ad + bc)/bd
- Always state restrictions: denominator ≠ 0.
- 务必注明限制条件：分母不得为零。
2. Linear Functions & Graphs | 线性函数与图像
A straight line is defined by y = mx + c, where m is the gradient and c is the y‑intercept. The gradient between two points (x₁, y₁) and (x₂, y₂) is given by:
直线由 y = mx + c 定义，其中 m 为斜率，c 为 y 轴截距。两点 (x₁, y₁) 与 (x₂, y₂) 间的斜率公式：
m = (y₂ – y₁) / (x₂ – x₁)
Parallel lines have equal gradients, while perpendicular lines satisfy m₁ × m₂ = –1. To find the equation of a line, use y – b = m(x – a) when you know a point (a, b) and the gradient m.
平行直线斜率相等；互相垂直的直线满足 m₁ × m₂ = –1。若已知一点 (a, b) 与斜率 m，可用 y – b = m(x – a) 求直线方程。
To sketch the graph, plot the y‑intercept c, then use the gradient to move right/left and up/down.
绘制图像时，先标出 y 截距 c，再利用斜率向右（或左）移动并向上（或下）移动描点。
3. Quadratic Equations & the Parabola | 二次方程与抛物线
A quadratic equation can be written as ax² + bx + c = 0, where a ≠ 0. Its solutions are found using the quadratic formula:
二次方程可写成 ax² + bx + c = 0（a ≠ 0）。求根公式为：
x = [–b ± √b² – 4ac] / (2a)
The discriminant, Δ = b² – 4ac, reveals the nature of the roots:
- Δ > 0 → two distinct real roots / 两个不等实根
- Δ = 0 → one repeated real root / 一个重根
- Δ < 0 → no real roots (complex) / 无实根
. If a > 0 it opens upward; if a < 0 it opens downward. The vertex (turning point) can be found by completing the square to the form y = a(x – h)² + k, giving (h, k).
y = ax² + bx + c 的图像为抛物线。a > 0 时开口向上，a < 0 时开口向下。顶点（转折点）可通过配方转化为 y = a(x – h)² + k 求得，坐标为 (h, k)。
4. Simultaneous Equations | 联立方程
Two linear equations can be solved by substitution or elimination. For elimination, multiply equations so that the coefficients of one variable become equal, then add or subtract to remove that variable.
两个线性方程可用代入法或消元法求解。消元法中，将方程乘以适当的倍数使某一变量的系数相等，再相加或相减以消去该变量。
When one equation is quadratic, substitute the linear expression into the non‑linear equation to form a single quadratic. Always check your solutions back in the original equations.
当一个方程为二次式时，将线性表达式代入非线性方程，得到一元二次方程。求出解后务必代回原方程检验。
Example: y = 2x + 1, x² + y = 7 → x² + 2x + 1 = 7
The solutions represent the intersection points of the corresponding graphs.
方程组的解对应相关图像的交点坐标。
5. Trigonometry in Right‑Angled Triangles | 直角三角形中的三角学
The three trigonometric ratios relate the angles and sides of a right‑angled triangle:
三个三角比将直角三角形的边与角联系起来：
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Common exact values must be memorised:
必须熟记常见角的精确值：
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
To solve a triangle, identify the known sides and the required ratio, then use the inverse trig function. Angles of elevation and depression are measured from the horizontal.
解直角三角形时，确定已知边和所需的三角比，再使用反三角函数。仰角和俯角均从水平线起算。
6. Non‑Right‑Angled Trigonometry | 非直角三角形三角学
For any triangle with sides a, b, c opposite angles A, B, C, the sine rule applies:
对任意三角形，边长 a, b, c 分别对角 A, B, C，正弦定理为：
a/sin A = b/sin B = c/sin C
Use the sine rule when you know two angles and a side (AAS) or two sides and a non‑included angle (SSA). The cosine rule links the three sides and one angle:
当已知两角一边 (AAS) 或两边及非夹角 (SSA) 时使用正弦定理。余弦定理将三边与一角联系起来：
a² = b² + c² – 2bc cos A
Use the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS). The area of a non‑right‑angled triangle is:
当已知两边及夹角 (SAS) 或三边 (SSS) 时应用余弦定理。非直角三角形面积公式：
Area = ½ ab sin C
7. Pythagoras & Circle Theorems | 勾股定理与圆定理
In any right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
在任何直角三角形中，斜边的平方等于两直角边的平方和。
c² = a² + b²
Key circle theorems include:
- The angle in a semicircle is 90°. / 半圆内的圆周角是 90°。
- The perpendicular from the centre to a chord bisects the chord. / 圆心到弦的垂线平分该弦。
- The tangent at any point is perpendicular to the radius at that point. / 圆的切线垂直于过切点的半径。
- Angles subtended by the same arc at the circumference are equal. / 同弧所对的圆周角相等。
These geometric facts often appear in multi‑step problems; always sketch a clear diagram and mark equal angles or radii.
这些几何定理常出现在多步问题中；务必画出清晰的示意图，标出相等的角或半径。
8. Vectors in 2D | 二维向量
A vector is a quantity with magnitude and direction, often written as a column vector or in component form:
向量是既有大小又有方向的量，通常表示为列向量或分量形式：
x = (x₁, x₂) or x₁ x₂
Addition and subtraction are performed component‑wise. Multiplication by a scalar k stretches or shrinks the vector, and if k is negative, reverses its direction.
加减法按分量分别运算。标量乘法 k 将向量伸缩，若 k 为负数则反向。
a + b = (a₁ + b₁, a₂ + b₂)
The magnitude (length) of vector v = (x, y) is |v| = √x² + y². A unit vector has magnitude 1; i and j are the standard unit vectors along the x‑ and y‑axes.
向量 v = (x, y) 的模长（长度）为 |v| = √x² + y²。单位向量的模为 1；i 与 j 分别是沿 x 轴与 y 轴的标准单位向量。
9. Statistics & Data Handling | 统计学与数据处理
The mean of a data set is the sum of all values divided by the number of values. For grouped data, use the midpoints of intervals.
数据的平均数等于所有数值之和除以数据个数。对于分组数据，使用各组的中点。
Mean = Σx / n, Grouped mean = Σ(fx) / Σf
The median is the middle value; for n items, it is the (n+1)/2‑th value when ordered. The interquartile range (IQR = Q₃ – Q₁) measures spread. The semi‑interquartile range is IQR/2.
中位数是中间值；将 n 个数据排序后，第 (n+1)/2 个即为中位数。四分位距 (IQR = Q₃ – Q₁) 衡量分散程度，半四分位距为 IQR/2。
Standard deviation is a more precise measure of spread, calculated for a population or a sample:
标准差是更精确的离散程度度量，分为总体与样本：
σ = √[ Σ(x – μ)² / N ] or s = √[ Σ(x – x̄)² / (n – 1) ]
Box plots are excellent for comparing distributions. Always comment on centre and spread.
箱线图非常适合比较分布。分析时务必对集中趋势和离散程度作评述。
10. Surds & Indices | 根式与指数
Surds are irrational roots that retain the root symbol. Simplifying surds often uses the fact that √ab = √a × √b and √a/b = √a / √b.
根式是保留根号的无理数。化简根式常用到 √ab = √a × √b 以及 √a/b = √a / √b。
Rationalising the denominator means removing a surd from the bottom of a fraction. For a single surd, multiply top and bottom by the surd; for a binomial, use the conjugate.
分母有理化即消除分母中的根号。对于单一根式，分子分母同乘该根式；对于二项式，则乘以其共轭式。
1/√a → √a/a ; 1/(√a + √b) → (√a – √b)/(a – b)
The laws of indices hold for all real exponents:
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ / aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1/aⁿ
- a1/n = ⁿ√a
指数律对所有实数指数成立：
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ / aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1/aⁿ
- a1/n = ⁿ√a
11. Sequences & Series | 数列与级数
In an arithmetic sequence, the difference d between consecutive terms is constant. The nth term is:
等差数列中相邻两项之差 d 为常数。第 n 项公式：
uₙ = a + (n – 1)d
where a is the first term. The sum of the first n terms can be found using:
其中 a 为首项。前 n 项和可用下式求得：
Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l)
with l being the last term.
l 为末项。
r. The nth term is uₙ = arⁿ⁻¹, and the sum of the first n terms (if r ≠ 1) is Sₙ = a(1 – rⁿ)/(1 – r).
等比数列的公比 r 恒定。第 n 项为 uₙ = arⁿ⁻¹，前 n 项和（当 r ≠ 1）为 Sₙ = a(1 – rⁿ)/(1 – r)。
Recognising whether a sequence is arithmetic, geometric, or neither is a common exam skill. Check the first differences for an arithmetic pattern and the ratios for a geometric one.
辨别数列是等差、等比还是其他类型是常见的考试技能。通过一次差判断等差，通过比值判断等比。
Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com
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