📚 Year 11 SQA Mathematics: Quick Reference Handbook of Formulas and Theorems | SQA 数学公式定理速查手册
This quick reference handbook summarises essential formulas and theorems for Year 11 SQA Mathematics, covering key topics such as algebra, geometry, trigonometry, and statistics. It is designed to help you quickly locate the facts you need for homework, classwork, and exam preparation. Keep this guide handy whenever you need a reliable reminder of the core content in the SQA National 5 course.
本速查手册总结了 Year 11 SQA 数学的基本公式和定理，涵盖代数、几何、三角学和统计等关键主题。它旨在帮助你快速查找作业、课堂学习以及备考所需的要点。在需要可靠地回顾 SQA National 5 课程核心内容时，随时查阅本指南。
1. Algebraic Expressions | 代数表达式
is process, aiming to write an expression as a product of its factors, e.g., ax² + bx + c = (px + q)(rx + s). Common factor must be taken out first when required.
去括号：将第一个括号中的每一项乘以第二个括号中的每一项。(a + b)(c + d) = ac + ad + bc + bd。因式分解是相反的过程，目的是将表达式写成其因式的乘积，例如 ax² + bx + c = (px + q)(rx + s)。需要时首先提取公因式。
: terms with the same variable and power can be added or subtracted. For instance, 5x + 3x – 2x = 6x.
化简表达式涉及合并同类项：具有相同变量和次数的项可以相加或相减。例如 5x + 3x – 2x = 6x。
2. Linear Equations | 线性方程
(slope) and c is the y-intercept (the point where the line crosses the y-axis). To solve a linear equation, aim to isolate the variable by performing inverse operations on both sides.
线性方程可以写成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴的交点）。解线性方程时，目标是通过两边进行逆运算来隔离变量。
the equation is y = c. For a vertical line, the equation is x = k, where k is a constant.
对于水平线，m = 0，方程为 y = c。对于垂直线，方程为 x = k，其中 k 是常数。
3. Quadratic Equations | 二次方程
ard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0. The discriminant Δ = b² – real roots; if Δ = 0, there is one repeated real root; if Δ < 0, there are no real roots.
二次方程的标准形式是 ax² + bx + c = 0，其中 a ≠ 0。判别式 Δ = b² – 4ac 决定根的性质：若 Δ > 0，有两个不相等的实根；若 Δ = 0，有一个重实根；若 Δ < 0，没有实根。
Quadratic Formula: x = (-b ± √b² – 4ac) / (2a)
二次公式：x = (-b ± √b² – 4ac) / (2a)
. Factorisation is often used when the quadratic can be expressed as (mx + n)(px + q) = 0.
配方法将 ax² + bx + c 写成 a(x + p)² + q。当二次式可以表示为 (mx + n)(px + q) = 0 时，常使用因式分解。
4. Geometry and Area | 几何与面积
l sides) × height. Area of a circle = π × radius². Circumference of a circle = 2 × π × radius or π × diameter.
关键面积公式：三角形面积 = ½ × 底 × 高。平行四边形面积 = 底 × 垂直高。梯形面积 = ½ × (平行边之和) × 高。圆的面积 = π × 半径²。圆的周长 = 2 × π × 半径 或 π × 直径。
tor = (θ/360) × 2πr. Sector area = (θ/360) × πr², where θ is the central angle in degrees.
组合图形可以分解为简单的图形；分别计算面积后相加。扇形的弧长 = (θ/360) × 2πr。扇形面积 = (θ/360) × πr²，其中 θ 是圆心角的度数。
5. Circle Properties | 圆的性质
The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc. Angles in the same segment of a circle are equal. The angle in a semicircle is a right angle (90°).
圆心角是同一弧所对圆周角的两倍。同弧上的圆周角相等。半圆内的圆周角是直角（90°）。
A tangent to a circle is perpendicular to the radius at the point of contact. Tangents from an external point to a circle are equal in length. The alternate segment theorem states that the angle between a tangent and a chord equals the angle in the alternate segment.
圆的切线垂直于切点处的半径。从圆外一点到该圆的两条切线长度相等。弦切角定理指出，切线与弦之间的夹角等于交替弦弧上的圆周角。
6. Trigonometry | 三角学
otenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Remember SOH CAH TOA.
在直角三角形中，三角比是：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记住 SOH CAH TOA。
Sine Rule: a / sin A = b / sin B = c / sin C
正弦定理：a / sin A = b / sin B = c / sin C
Cosine Rule: a² = b² + c² – 2bc cos A
余弦定理：a² = b² + c² – 2bc cos A
trigonometry: Area = ½ ab sin C, where a and b are two sides and C is the included angle.
用三角学求三角形面积：面积 = ½ ab sin C，其中 a 和 b 是两条边，C 是夹角。
7. Pythagoras' Theorem | 毕达哥拉斯定理
This theorem is used to find missing side lengths and to prove whether a triangle is right-angled.
在任意直角三角形中，斜边的平方等于另外两条边的平方和：a² + b² = c²，其中 c 是斜边。该定理用于求缺失的边长及印证三角形是否为直角三角形。
nce formula between two points (x₁, y₁) and (x₂, y₂) is derived from Pythagoras: d = √[(x₂ – x₁)² + (y₂ – y₁)²]. This is essential in coordinate geometry.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离公式来自毕达哥拉斯定理：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。这在坐标几何中很重要。
8. Statistics and Probability | 统计与概率
es). Median = middle value when ordered. Mode = most frequent value. Range = highest value – uartiles divide data into four equal parts; the interquartile range (IQR) = upper quartile – lower quartile.
集中趋势的度量：平均数 = (数值总和) ÷ (数值个数)。中位数 = 排序后的中间值。众数 = 出现最频繁的值。范围 = 最大值 – 最小值。四分位数将数据分成四等份；四分位距 (IQR) = 上四分位数 – 下四分位数。
vourable outcomes / total number of possible outcomes. The probability of an event not occurring is 1 – P(event). Tree diagrams and two-way tables help list outcomes for combined events.
概率：P(事件) = 有利结果的数量 / 所有可能结果的总数。事件不发生的概率是 1 – P(事件)。树形图和双向表有助于列出复合事件的结果。
9. Volume and Surface Area | 体积与表面积
Volume of a prism = area of cross‑ r²h. Volume of a pyramid = ⅓ × base area × vertical height. Volume of a sphere = (4/3)πr³.
棱柱的体积 = 横截面积 × 长度。圆柱体：V = πr²h。圆锥体的体积 = ⅓πr²h。棱锥的体积 = ⅓ × 底面积 × 垂直高。球体的体积 = (4/3)πr³。
height. Surface area of a sphere = 4πr². For composite solids, treat each part separately.
圆柱的表面积 = 2πr² + 2πrh。圆锥的表面积（含底面）= πrl + πr²，其中 l 是斜高。球的表面积 = 4πr²。对于组合体，将各部分分别处理。
10. Gradient and Straight Line | 斜率与直线方程
The gradient between two points (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁). The equation of a straight line can be written as y = mx + c, or in point‑gradient form: y – b = m(x – a).
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率是 m = (y₂ – y₁) / (x₂ – x₁)。直线方程可以写为 y = mx + c，或点斜式：y – b = m(x – a)。
l lines have equal gradients: m₁ = m₂. Perpendicular lines have gradients whose product is –1: m₁ × m₂ = –1. The intercepts on the axes can be found by setting x = 0 or y = 0.
平行线具有相等的斜率：m₁ = m₂。互相垂直的直线斜率的乘积为 –1：m₁ × m₂ = –1。与坐标轴的截距可通过设 x = 0 或 y = 0 求得。
11. Simultaneous Equations | 联立方程组
Two linear equations in two unknowns can be solved by substitution or elimination. Substitution involves rearranging one equation to express one variable in terms of the other and substituting into the second equation. Elimination involves adding or subtracting equations to remove one variable.
两个未知数的两个线性方程组可以通过代入法或消元法求解。代入法即重组一个方程，将一个变量用另一个变量表示，然后代入第二个方程。消元法是对方程进行加减以消去一个变量。
Graphical solution: the point of intersection of the two lines gives the solution. Check your solutions by substituting back into both original equations.
图解：两条直线的交点即为解。将解代回原方程以检验。
12. Indices and Surds | 指数与根式
Laws of indices: am × an = am+n. am ÷ an = am–n. (am)n = amn. a0 = 1 (a ≠ 0). a–n = 1 / an. a1/n = n√a.
指数运算法则：am × an = am+n。am ÷ an = am–n。(am)n = amn。a0 = 1（a ≠ 0）。a–n = 1 / an。a1/n = n√a。
Surds can be simplified by looking for square factors: √ab = √a × √b. To rationalise a denominator such as 1/√a, multiply top and bottom by √a. For expressions like 1/(a + √b), multiply by the conjugate (a – √b).
根式可以通过寻找平方因子来化简：√ab = √a × √b。要使分母如 1/√a 有理化，分子分母同乘以 √a。对于如 1/(a + √b) 的表达式，乘以共轭式 (a – √b)。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图