📚 Homework 7C Prelims: Core Mathematical Skills for Pre-Exam Success | 作业 7C 预考：考前数学核心技能
The Homework 7C Prelims paper is designed to test the foundational skills that underpin success in advanced mathematics. This article revisits the key algebraic, trigonometric, calculus, and coordinate geometry techniques that frequently appear in preliminary examinations. By working through the core methods and common pitfalls, you can build both accuracy and confidence before the final assessment.
作业 7C 预考试卷旨在考查支撑高等数学成功的基础技能。本文重新梳理代数、三角、微积分和坐标几何中经常出现在预考中的核心技巧。通过攻克核心方法和常见易错点，你可以在最终评估前提升准确度与信心。
1. Algebraic Manipulation and Simplification | 代数运算与化简
When simplifying algebraic fractions, factorise the numerator and denominator first. For example, (x² – 4)/(x – 2) becomes (x + 2) after cancelling the common factor.
化简代数分式时，应先对分子和分母进行因式分解。例如 (x² – 4)/(x – 2) 约去公因式后化为 (x + 2)。
Remember that only factors can be cancelled, never terms. The expression (x + 3)/x cannot be simplified to 3 because x is a term in the numerator, not a factor.
记住只有因式才能约分，项不能约分。表达式 (x + 3)/x 不能化简为 3，因为 x 是分子中的一项而非因式。
When expanding brackets, write each product separately before collecting like terms. For (x + 2)(x – 5), expand to x² – 5x + 2x – 10, then simplify to x² – 3x – 10.
展开括号时，先将每个乘积分别写出，再合并同类项。对于 (x + 2)(x – 5)，展开为 x² – 5x + 2x – 10，再化简为 x² – 3x – 10。
Always state excluded values for rational expressions. In the first example, x ≠ 2 must be noted because the original denominator would be zero.
务必注明有理式的排除值。在第一个例子中，必须注明 x ≠ 2，因为原分母会为零。
a² – b² = (a – b)(a + b)
2. Coordinate Geometry and Straight Lines | 坐标几何与直线
The gradient of a line through points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ – y₁)/(x₂ – x₁). The equation y – y₁ = m(x – x₁) is often faster than y = mx + c when a point and gradient are known.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ – y₁)/(x₂ – x₁)。当已知一点和斜率时，使用 y – y₁ = m(x – x₁) 往往比 y = mx + c 更快。
Parallel lines have equal gradients, while perpendicular lines satisfy m₁m₂ = -1. In exam questions, check whether the line is given in gradient-intercept form or general form before deciding the approach.
平行直线斜率相等，而垂直直线满足 m₁m₂ = -1。在考试题中，先判断直线是以斜截式还是一般式给出，再决定方法。
The distance between two points is found using the Pythagorean theorem: d = √[(x₂ – x₁)² + (y₂ – y₁)²]. This formula also appears in circle questions when finding the radius.
两点间距离使用勾股定理求得：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。该公式也出现在求圆的半径的题目中。
For circles, the standard form is (x – a)² + (y – b)² = r², where (a, b) is the centre and r is the radius. Complete the square to convert a general circle equation into this form.
对于圆，标准形式为 (x – a)² + (y – b)² = r²，其中 (a, b) 是圆心，r 是半径。将一般式圆方程化为该形式时使用配方法。
m = (y₂ – y₁)/(x₂ – x₁)
3. Functions and Transformations | 函数与图像变换
A function maps each input x to exactly one output f(x). To find the inverse f⁻¹(x), swap x and y in the equation y = f(x) and solve for y, then check the domain of the inverse.
函数将每个输入 x 映射到唯一输出 f(x)。求逆函数 f⁻¹(x) 时，交换 y = f(x) 中的 x 和 y，再解出 y，然后检查逆函数的定义域。
The domain is the set of allowed input values, and the range is the set of possible output values. For f(x) = √(x – 1), the domain is x ≥ 1 and the range is f(x) ≥ 0.
定义域是允许的输入值集合，值域是可能的输出值集合。对于 f(x) = √(x – 1)，定义域为 x ≥ 1，值域为 f(x) ≥ 0。
Transformations such as f(x + a), f(x) + a, f(ax), and af(x) must be applied in the correct order. A horizontal shift is inside the bracket, while a vertical shift is outside.
f(x + a)、f(x) + a、f(ax) 和 af(x) 等变换必须按正确顺序进行。水平平移发生在括号内，而垂直平移发生在括号外。
For composite functions, work from the inside out. If f(x) = 2x and g(x) = x + 3, then fg(x) = f(x + 3) = 2(x + 3) = 2x + 6.
对于复合函数，从内到外进行。若 f(x) = 2x 且 g(x) = x + 3，则 fg(x) = f(x + 3) = 2(x + 3) = 2x + 6。
4. Trigonometry Fundamentals | 三角学基础
The exact values of sin, cos, and tan for 30°, 45°, and 60° are frequently tested. For example, sin 30° = 1/2, cos 45° = √2/2, and tan 60° = √3.
30°、45° 和 60° 的 sin、cos、tan 精确值经常考。例如 sin 30° = 1/2，cos 45° = √2/2，tan 60° = √3。
When solving trigonometric equations in a given interval, always find the principal value first using the inverse function, then use the symmetry of the unit circle to locate all solutions.
在给定区间内解三角方程时，先用反函数求出主值，然后利用单位圆的对称性找到所有解。
The identity sin²θ + cos²θ = 1 is useful for simplifying expressions and for proving other identities. It also leads to tan²θ + 1 = sec²θ and cot²θ + 1 = cosec²θ.
恒等式 sin²θ + cos²θ = 1 对化简表达式和证明其他恒等式都很有用。它还可推出 tan²θ + 1 = sec²θ 和 cot²θ + 1 = cosec²θ。
Remember that trigonometric functions are periodic: sin and cos repeat every 360° (or 2π radians), while tan repeats every 180° (or π radians). Always consider the given interval when listing solutions.
记住三角函数具有周期性：sin 和 cos 每 360°（或 2π 弧度）重复一次，而 tan 每 180°（或 π 弧度）重复一次。列出解时务必考虑给定区间。
sin²θ + cos²θ = 1
5. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d, so the nth term is uₙ = a + (n – 1)d. The sum of the first n terms is Sₙ = n/2[2a + (n – 1)d] or Sₙ = n/2(a + l).
等差数列具有常数公差 d，因此第 n 项为 uₙ = a + (n – 1)d。前 n 项和为 Sₙ = n/2[2a + (n – 1)d] 或 Sₙ = n/2(a + l)。
A geometric sequence has a constant ratio r, with nth term uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 – rⁿ)/(1 – r) when r ≠ 1.
等比数列具有常数公比 r，第 n 项为 uₙ = arⁿ⁻¹。当 r ≠ 1 时，前 n 项和为 Sₙ = a(1 – rⁿ)/(1 – r)。
The sum to infinity exists only when |r| < 1 and is given by S∞ = a/(1 - r). This formula is often used to find the limit of a convergent geometric series.
无穷级数只有在 |r| < 1 时收敛，其和为 S∞ = a/(1 - r)。该公式常用于求解收敛等比级数的极限。
Sigma notation Σ is a compact way to write series. For example, Σ from k = 1 to n of k means 1 + 2 + 3 + … + n, which equals n(n + 1)/2.
求和符号 Σ 是级数的紧凑写法。例如 Σ 从 k = 1 到 n 的 k 表示 1 + 2 + 3 + … + n，其和为 n(n + 1)/2。
S∞ = a/(1 – r), |r| < 1
6. Differentiation Rules and Applications | 微分法则与应用
The power rule states that if y = xⁿ, then dy/dx = nxⁿ⁻¹. This applies to polynomial terms separately, and constant terms differentiate to zero.
幂法则指出，若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。这适用于多项式中的每一项，常数项求导为零。
The chain rule is used for composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). For example, the derivative of (2x + 1)³ is 6(2x + 1)².
链式法则用于复合函数：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。例如 (2x + 1)³ 的导数为 6(2x + 1)²。
For tangents and normals, first find dy/dx at the point of contact. The tangent gradient is m, while the normal gradient is -1/m.
对于切线和法线，先求接触点处的 dy/dx。切线斜率为 m，法线斜率为 -1/m。
Stationary points occur where dy/dx = 0. Use the second derivative d²y/dx² or the sign change of dy/dx to classify a maximum, minimum, or point of inflection.
驻点出现在 dy/dx = 0 处。利用二阶导数 d²y/dx² 或 dy/dx 的符号变化来判断极大值、极小值或拐点。
d/dx [xⁿ] = nxⁿ⁻¹
7. Integration Basics | 积分基础
Integration is the reverse of differentiation. For xⁿ with n ≠ -1, ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, where C is the constant of integration.
积分是微分的逆运算。对于 n ≠ -1 的 xⁿ，∫ xⁿ dx = xⁿ⁺¹
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