CCEA Further Maths Year 11 Formula & Theorem Quick Reference | CCEA 进阶数学 Year 11 公式定理速查手册

📚 CCEA Further Maths Year 11 Formula & Theorem Quick Reference | CCEA 进阶数学 Year 11 公式定理速查手册

A concise handbook of essential formulas and theorems from the CCEA Further Mathematics Year 11 syllabus, ideal for quick revision and reference before assessments. Work through it section by section, cover the formulas and try to reproduce each statement from memory, then use the notes to check the exact conditions under which every result is valid.

一本涵盖 CCEA 进阶数学 Year 11 核心公式与定理的速查手册，适合在测试前快速复习与查阅。建议按小节逐条过一遍：遮住公式，尝试凭记忆复述，再用说明文字核对每个结论成立的条件与适用范围。

1. Algebraic Expressions | 代数表达式

a(b + c) = ab + ac

The distributive law removes brackets by multiplying the outside factor by every term inside, and it is the first step in almost every expansion question.

分配律：将括号外的因子分别乘入括号内的每一项，这是几乎所有展开题的第一步。

(a + b)² = a² + 2ab + b²

The perfect square of a sum: the middle term 2ab is the most commonly forgotten part of this expansion.

完全平方和公式：展开二项和的平方，中间项 2ab 是最容易被漏掉的部分。

(a − b)² = a² − 2ab + b²

The perfect square of a difference, identical to the previous identity except for the sign of the middle term.

完全平方差公式：与上一式仅中间项符号不同。

a² − b² = (a − b)(a + b)

The difference of two squares factorises whenever you see two terms separated by a minus sign, and it is also the standard trick for rationalising denominators.

平方差公式：只要看到两项相减，就优先考虑因式分解，它同时也是分母有理化的标准技巧。

a³ + b³ = (a + b)(a² − ab + b²)

Factorisation of a sum of two cubes. Note the sign pattern: plus outside, minus in the middle bracket.

立方和因式分解。注意符号规律：外层为加号，中间括号内为减号。

a³ − b³ = (a − b)(a² + ab + b²)

Factorisation of a difference of two cubes, following the same sign pattern in reverse.

立方差因式分解，符号规律与上式相反。

Remainder theorem: remainder = f(a) when f(x) is divided by (x − a)

n, and the factor theorem states that f(a) = 0 if and only if (x − a) is a factor of f(x).

余数定理：f(x) 除以 (x − a) 的余数等于 f(a)；因式定理：f(a) = 0 当且仅当 (x − a) 是 f(x) 的因式。两者都只需代入即可判断。

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2. Quadratic Functions | 二次函数

f(x) = ax² + bx + c (a ≠ 0)

This general form describes a parabola that opens upwards when a > 0 and downwards when a < 0; the constant c is the y-intercept.

二次函数一般式：当 a > 0 时抛物线开口向上，a < 0 时开口向下，常数项 c 即 y 轴截距。

Δ = b² − 4ac

wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.

判别式决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个实根（重根），Δ < 0 无实根。

x = (−b ± √b² − 4ac) / (2a)

x² + bx + c = 0 for any values of the coefficients, including those that do not factorise.

求根公式用于解 ax² + bx + c = 0，适用于所有系数情形，尤其是无法因式分解的方程。

Vertex: x = −b/(2a), y = f(−b/(2a))

function as y = a(x − h)² + k, so the vertex is (h, k) and the line of symmetry is x = h.

配方法可将函数写成 y = a(x − h)² + k，顶点为 (h, k)，对称轴为 x = h。

Sum of roots = −b/a, product of roots = c/a

These two relations let you build a quadratic from its roots, or find unknown coefficients without solving the equation.

根与系数的关系：可由两根构造二次方程，也可在不求解的情况下求出未知系数。

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3. Equations and Inequalities | 方程与不等式

ax + b = 0 → x = −b/a (a ≠ 0)

A linear equation in one unknown is solved by isolating x, and the same rearrangement is used when the subject of a formula has to be changed.

一元一次方程通过移项、合并同类项求出 x；换主元（改变公式主项）时使用的也是同一套变形方法。

| ax + b | = c → ax + b = ±c (c ≥ 0)

An absolute value equation splits into two separate linear equations, and if c < 0 the equation has no solution at all.

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绝对值方程拆分为两个线性方程；若 c < 0，则原方程无解。

(x − p)(x − q) < 0 ⇔ p < x < q (assuming p < q)

When the product is negative the solution lies strictly between the two roots, giving a closed interval of values.

当乘积小于零时，解集位于两根之间，写成区间形式即为 (p, q)。

(x − p)(x − q) > 0 ⇔ x < p or x > q

When the product is positive the solution lies outside the roots, so the answer is the union of two separate intervals.

当乘积大于零时，解集位于两根之外，答案是两个区间的并集。

For a general quadratic inequality ax² + bx + c > 0, first solve the corresponding equation to find the critical values, mark them on a number line, then test one value in each interval to determine the sign.

对于一般二次不等式 ax² + bx + c > 0：先解对应方程求出临界值，将其标在数轴上，再在每个区间内取一个测试点判断符号，最后写出解集。

Remember that multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign; in a fractional inequality, move everything to one side rather than cross-multiplying blindly.

切记：不等式两边同乘或同除一个负数时，不等号方向要反转；分式不等式应先把所有项移到同一边通分，不要盲目交叉相乘。

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4. Indices and Surds | 指数与根式

aᵐ × aⁿ = aᵐ⁺ⁿ

When powers with the same base are multiplied, the indices are added while the base stays unchanged.

同底数幂相乘，底数不变，指数相加。

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

When powers with the same base are divided, the indices are subtracted.

同底数幂相除，底数不变，指数相减。

(aᵐ)ⁿ = aᵐⁿ

A power raised to another power multiplies the indices, a rule that also applies to the outside of a bracket containing a product.

幂的乘方，指数相乘；括号内为乘积时，指数同样分配给每个因子。

a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ

Any non-zero base raised to the power zero equals one, and a negative index signals a reciprocal.

任何非零数的零次幂等于 1；负指数表示取倒数。

a^1/n = ⁿ√a, a^m/n = ⁿ√aᵐ

Fractional indices convert directly into roots, with the denominator giving the root and the numerator giving the power.

分数指数与根式互化：分母为根指数，分子为幂指数。

√a × √b = √ab, √a ÷ √b = √a/b (b ≠ 0)

Surds may be multiplied and divided under a single root sign, provided the values inside stay non-negative.

根式的乘法与除法可合并到同一个根号下，前提是根号内的值非负。

1/√a = √a/a, 1/(a + √b) = (a − √b)/(a² − b)

Rationalising the denominator removes the surd from the bottom of a fraction by multiplying numerator and denominator by the surd or by its conjugate.

分母有理化：分子分母同乘根式本身或其共轭表达式，使分母不再含根式。

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5. Logarithms | 对数

If aˣ = b then logₐ b = x (a > 0, a ≠ 1, b > 0)

A logarithm answers the question of what power the base must be raised to in order to produce the given number, so it is the inverse of an exponential.

对数的本质是回答“底数要取多少次幂才能得到该数”，因此它是指数运算的逆运算。

logₐ(mn) = logₐ m + logₐ n

The logarithm of a product equals the sum of the logarithms, which converts multiplication into addition.

积的对数等于对数的和，即把乘法转化为加法。

logₐ(m/n) = logₐ m − logₐ n

The logarithm of a quotient equals the difference of the logarithms.

商的对数等于对数的差。

logₐ(mᵏ) = k logₐ m

A power inside a logarithm comes down as a multiplying factor, which is the key step in solving exponential equations.

真数的幂可以提到对数

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