📚 Year 11 SQA Maths: Quick Reference Formula and Theorem Handbook | Year 11 SQA 数学：公式定理速查手册
This quick reference handbook brings together essential formulas, theorems and relationships you will need for Year 11 SQA Mathematics. Whether you are working towards National 5 or beginning Higher, the summaries below cover algebra, geometry, trigonometry, statistics, vectors and calculus core content. Use it as a revision checklist, a memory booster or a last‑minute glance before assessments.
这本速查手册汇集了 Year 11 SQA 数学所需的核心公式、定理与关系。无论你正在准备 National 5 还是刚刚接触 Higher 内容，下面的总结都涵盖了代数、几何、三角学、统计、向量以及微积分的基础知识。你可以把它当作复习清单、记忆强化器或考前快速浏览的帮手。
1. Algebraic Expressions and Formulae | 代数表达式与公式
Expanding brackets relies on the distributive law. For two binomials, multiply each term in the first bracket by each term in the second bracket and collect like terms.
去括号依赖分配律。对于两个二项式相乘，将第一个括号中的每一项与第二个括号中的每一项相乘，然后合并同类项。
(a + b)(c + d) = ac + ad + bc + bd
A common factor can be taken out to simplify expressions. Always look for the highest common factor first.
可以提取公因式来化简表达式。务必先寻找最大公因式。
ax + ay = a(x + y)
The difference of two squares is a special factorisation pattern that appears frequently in quadratics and surds.
平方差公式是一种特殊的因式分解模式，在二次式和根式中经常出现。
a² – b² = (a – b)(a + b)
Completing the square transforms a quadratic expression into vertex form. It is useful for finding maximum or minimum values and for solving equations.
配方法将二次表达式转化为顶点式，这对求最大值或最小值以及解方程都很有用。
x² + bx + c = (x + b/2)² – (b/2)² + c
2. Linear Equations and Graphs | 线性方程与图像
The gradient‑intercept form of a straight line gives the gradient m and the y‑intercept c directly. The line crosses the y‑axis at (0, c).
直线的斜截式直接给出了斜率 m 和 y 轴截距 c。该直线与 y 轴相交于点 (0, c)。
y = mx + c
The gradient between two points (x₁, y₁) and (x₂, y₂) measures the rate of vertical change relative to horizontal change. Label the points carefully to avoid sign errors.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率衡量了垂直变化相对于水平变化的速率。标明点时要仔细，避免符号错误。
m = (y₂ – y₁) / (x₂ – x₁)
Understanding the conditions for parallel and perpendicular lines saves time when solving geometry problems or finding equations of tangents.
理解平行与垂直的条件可以在解几何问题或求切线方程时节省时间。
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = –1
3. Quadratic Equations and Functions | 二次方程与函数
A quadratic function in standard form produces a parabola. The sign of a determines whether the parabola opens upwards or downwards.
标准形式的二次函数生成一条抛物线。a 的符号决定了抛物线开口向上还是向下。
y = ax² + bx + c
The quadratic formula solves ax² + bx + c = 0 for any real coefficients. The expression under the square root, b² – 4ac, is called the discriminant.
二次公式可解任意实系数的方程 ax² + bx + c = 0。平方根下的表达式 b² – 4ac 称为判别式。
x = [–b ± √(b² – 4ac)] / (2a)
The discriminant reveals the nature of the roots without solving the equation fully. A positive discriminant gives two real roots, zero gives one repeated root, and a negative discriminant gives no real roots.
判别式无需完全解方程即可揭示根的性质。正判别式给出两个实数根，零给出一个重根，负判别式则无实数根。
Δ = b² – 4ac
4. Trigonometry in Right-Angled Triangles | 直角三角形三角学
SOH CAH TOA relates the acute angles of a right‑angled triangle to its side ratios. Choose the ratio that involves the known side and the unknown side.
SOH CAH TOA 将直角三角形的锐角与其边长比联系起来。选择涉及已知边和未知边的比值。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Certain angles give exact trigonometric values that appear frequently in non‑calculator papers. Memorising these values can speed up your working.
一些特殊角具有精确的三角值，经常出现在不允许使用计算器的试卷中。记住这些值可以加快解题速度。
- sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2
- cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2
- tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
Pythagoras’ theorem connects the three sides of any right‑angled triangle. It is also a key tool in coordinate geometry for finding distances.
勾股定理将任意直角三角形的三条边联系起来。它也是坐标几何中求距离的关键工具。
a² + b² = c² (where c is the hypotenuse)
5. Sine and Cosine Rules | 正弦定理与余弦定理
The sine rule is used for non‑right‑angled triangles when you know two angles and one side, or two sides and a non‑included angle. Set up the ratios with care.
正弦定理用于非直角三角形，已知两角一边或两边及一个非夹角的情况。建立比例时要仔细。
a / sin A = b / sin B = c / sin C
The cosine rule generalises Pythagoras’ theorem. Use it when you know two sides and the included angle, or all three sides and need an angle.
余弦定理是勾股定理的推广。当已知两边及其夹角，或已知三边求角时使用。
a² = b² + c² – 2bc cos A
cos A = (b² + c² – a²) / (2bc)
The area of any triangle can be found using the sine of an included angle. This formula is particularly useful when the perpendicular height is not given.
任意三角形的面积都可以用夹角的正弦值求得。当未给出垂直高度时，这个公式尤其有用。
Area = ½ ab sin C
6. Circle Geometry | 圆几何
Circumference and area formulas are the building blocks for arc lengths and sector areas. Always check whether the question expects an exact answer in terms of π or a rounded decimal.
周长和面积公式是计算弧长和扇形面积的基础。务必检查题目要求是以 π 表示精确值还是四舍五入的小数。
C = πd = 2πr
A = πr²
Arc length and sector area are proportional to the central angle. Work with fractions of 360° for degree measure.
弧长和扇形面积与圆心角成正比。使用角度制时，按 360° 的比例计算。
Arc length = θ/360 × 2πr
Sector area = θ/360 × πr²
Two circle theorems appear regularly in angle problems. A tangent meets a radius at 90°, and the angle at the centre is twice the angle at the circumference standing on the same arc.
在角度问题中经常出现两个圆定理。切线与半径相交成 90°，圆心角是同弧上圆周角的两倍。
Tangent ⊥ radius
∠ at centre = 2 × ∠ at circumference
7. Statistics and Probability | 统计与概率
Measures of central tendency and spread summarise data sets effectively. The mean is sensitive to extreme values, while the median and interquartile range are more robust.
集中趋势和离散程度的度量能有效概括数据集。均值易受极端值影响，而中位数和四分位距则更加稳健。
- Mean = (sum of all values) ÷ (number of values)
- Range = maximum – minimum
- Interquartile range (IQR) = Q₃ – Q₁
Standard deviation measures the average distance of data points from the mean. The SQA formula booklet provides both sample and population versions; use the sample formula unless told otherwise.
标准差衡量数据点与均值的平均距离。SQA 公式手册提供了样本和总体两种公式；除非另有说明，一般使用样本公式。
s = √[ Σ(x – x̄)² / (n – 1) ]
Probability rules help quantify chance. For mutually exclusive events, simply add probabilities; for independent events, multiply them.
概率法则有助于量化偶然性。对于互斥事件，直接将概率相加；对于独立事件，将概率相乘。
P(A or B) = P(A) + P(B) – P(A and B)
P(A and B) = P(A) × P(B) for independent events
8. Vectors in 2D | 二维向量
A vector is described by its components in the x and y directions. Adding or subtracting vectors is done component‑wise.
向量由其沿 x 方向和 y 方向的分量描述。向量的加减按分量进行。
u = (x, y) or u = (x)⁄(y)
The magnitude of a vector is found using Pythagoras’ theorem. Multiplying a vector by a scalar stretches or shrinks it without changing its direction (unless the scalar is negative).
向量的大小用勾股定理求得。标量乘法会拉伸或缩短向量但不改变其方向（除非标量为负）。
|u| = √(x² + y²)
k (x, y) = (kx, ky)
9. Indices and Surds | 指数与根式
The laws of indices allow expressions with powers to be manipulated efficiently. They are essential when working with polynomials, exponentials and algebraic fractions.
指数法则可以高效地处理含有幂的表达式。在处理多项式、指数函数和代数分式时它们必不可少。
aᵐ × aⁿ = aᵐ⁺ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1
a⁻ⁿ = 1 / aⁿ
a¹/ⁿ = ⁿ√a
Surds are irrational numbers in root form. Simplify surds by identifying square factors and rationalise denominators where required.
根式是无理数的根号形式。通过找出平方因子来化简根式，并在需要时分母有理化。
√a × √b = √(ab)
√a / √b = √(a/b)
Rationalising: 1/√a → √a / a
10. Differentiation Basics | 微分初步
The derivative of xⁿ follows a simple power rule. This rule underpins gradient calculations for curves and is the first step in optimisation problems.
xⁿ 的导数遵循简单的幂法则。该法则是曲线斜率计算的基础，也是最优化问题的第一步。
If y = xⁿ, then dy/dx = n xⁿ⁻¹ (for all real n)
Stationary points occur where the derivative equals zero. A sign diagram of the derivative tells you whether the point is a local maximum, local minimum or a point of inflection.
驻点出现在导数为零的位置。通过导数的符号表可以判断该点是局部极大值、局部极小值还是拐点。
For stationary points: dy/dx = 0
The derivative at a point gives the gradient of the tangent to the curve there. Combine this with y – y₁ = m(x – x₁) to write the equation of the tangent.
函数在某点的导数给出该点处切线的斜率。结合点斜式 y – y₁ = m(x – x₁) 即可写出切线方程。
11. Integration Basics | 积分初步
Integration reverses differentiation. The indefinite integral of a polynomial term raises the power by one and divides by the new power, then adds the constant of integration C.
积分是微分的逆运算。多项式项的不定积分将指数加 1 后除以新的指数，并加上积分常数 C。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ –1)
Definite integration computes the exact area between a curve and the x‑axis over an interval. Substitute the upper and lower limits and subtract.
定积分计算区间内曲线与 x 轴之间的精确面积。代入上限和下限并相减。
Area = ∫ₐᵇ f(x) dx = F(b) – F(a)
Always sketch the graph first if the function crosses the x‑axis; split the integral into separate parts to ensure all areas are treated as positive contributions.
如果函数穿过 x 轴，务必先画草图；将积分分段，确保所有面积均作为正值处理。
12. Straight Line and Simultaneous Equations Review | 直线与联立方程回顾
The general equation of a straight line can be written in various forms. The form Ax + By + C = 0 is often useful for determining intersections with axes.
直线的一般方程可以写成多种形式。Ax + By + C = 0 的形式常用于确定与坐标轴的交点。
Ax + By + C = 0
To solve a pair of simultaneous linear equations, you can use elimination (adding or subtracting equations to remove one variable) or substitution (rearranging one equation for one variable and plugging it into the other). Choose the method that keeps the numbers clean.
解二元一次联立方程组，可以使用消元法（将方程相加或相减以消去一个变量）或代入法（将一个方程变形为一个变量再代入另一个）。选择能让数字保持简洁的方法。
- Elimination: 2x + y = 7 and x – y = 2 → Add: 3x = 9 → x = 3, then y = 1.
- Substitution: from y = 2x – 1, substitute into 3x + 2y = 12.
Always check your solution by substituting both values back into the original equations. This catches arithmetic slips and ensures accuracy in exams.
务必将解代回原方程检验。这能发现算术错误，确保考试中的准确性。
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