Math Exam Prep: Core Problem-Solving Techniques & Common Mistakes | 数学备考：核心做题技巧与易错点分析

📚 Math Exam Prep: Core Problem-Solving Techniques & Common Mistakes | 数学备考：核心做题技巧与易错点分析

Mathematics exams reward not only knowledge but also the ability to apply that knowledge accurately under time pressure. Strong problem-solving techniques help you avoid careless errors, while knowledge of common pitfalls allows you to turn frequent mistakes into marks.

数学考试不仅考查知识，更考查在有限时间内准确运用知识的能力。掌握核心做题技巧能帮助你避免粗心错误，而了解常见易错点则能让你把失分点转化为得分点。

1. Read the Question Carefully | 仔细审题

Before writing anything, underline key terms such as ‘exact value’, ‘integer’, ‘degree’ or ‘radian’, and ‘non-calculator’. These words change the method completely.

动笔前，先圈出题目中的关键词，例如“精确值”“整数”“角度制”或“弧度制”“不可用计算器”。这些词会完全改变解题方法。

For example, ‘solve exactly’ means you need a surd or fraction, not a decimal. ‘Show that’ requires a logical chain, not just an answer.

例如，“精确求解”意味着需要保留根号或分数，而不是小数；“证明”需要逻辑链条，而不只是给出答案。

When you finish, check that your answer matches the command words and any domain restrictions.

完成后，检查答案是否符合指令词以及定义域限制。

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2. Master Algebraic Manipulation | 掌握代数变形

A large share of exam marks depends on fluent algebraic manipulation. Practice expanding, factorising, and simplifying expressions until they become automatic.

考试中很大一部分分数取决于熟练的代数变形能力。反复练习展开、因式分解和化简，直到这些操作成为本能。

Many students lose marks when handling negative indices and fractional powers. Remember that x⁻ⁿ = 1 / xⁿ and x^(m/n) = (ⁿ√x)ᵐ.

许多学生在处理负指数和分数幂时失分。记住 x⁻ⁿ = 1 / xⁿ 以及 x^(m/n) = (ⁿ√x)ᵐ。

Always write out each step neatly. Skipping steps in algebra is the most common source of sign errors.

每一步都要工整地写出。跳步是符号错误最常见的来源。

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3. Quadratics and the Discriminant | 二次方程与判别式

Quadratics appear throughout the syllabus. Ensure you can solve them by factorising, completing the square, and using the quadratic formula.

二次方程贯穿整个考纲。务必掌握因式分解、配方法和求根公式三种解法。

For an equation ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. If Δ > 0 there are two distinct roots; Δ = 0 gives one repeated root; Δ < 0 means no real roots.

对于方程 ax² + bx + c = 0，判别式为 Δ = b² − 4ac。若 Δ > 0，有两个不同实根；Δ = 0，有一个重根；Δ < 0，则没有实根。

A common mistake is to forget that when the coefficient of x² is negative, the graph opens downwards. This affects inequality signs.

一个常见错误是忘记当 x² 的系数为负时，抛物线开口向下，这会影响不等号方向。

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4. Functions and Transformations | 函数与图像变换

When working with functions, always state the domain and range. Many marks are lost for omitting them.

处理函数时，一定要写明定义域和值域。很多分数因为遗漏这两项而丢失。

Understand the difference between f(x) + a (vertical translation) and f(x + a) (horizontal translation). The horizontal shift is in the opposite direction to the sign.

理解 f(x) + a（上下平移）与 f(x + a)（左右平移）的区别。水平平移的方向与符号相反。

To find the inverse function, swap x and y, then rearrange. After finding the inverse, check that the range of the inverse equals the domain of the original function.

求反函数时，交换 x 与 y，再整理方程。求出后，检查反函数的值域是否等于原函数的定义域。

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5. Differentiation: Chain, Product, Quotient | 微分：链式、乘积、商法则

Memorise the three main differentiation rules. The chain rule states dy/dx = dy/du × du/dx. Use it for composite functions.

牢记三个主要微分法则。链式法则为 dy/dx = dy/du × du/dx，用于复合函数。

The product rule states d/dx(uv) = u dv/dx + v du/dx. The quotient rule states d/dx(u/v) = (v du/dx − u dv/dx) / v².

乘积法则为 d/dx(uv) = u dv/dx + v du/dx。商法则为 d/dx(u/v) = (v du/dx − u dv/dx) / v²。

A frequent error is differentiating each factor separately instead of applying the product rule. Always identify which rule fits before differentiating.

常见错误是把乘积的每一项分开求导，而不是应用积的法则。求导前先判断适用哪种法则。

For stationary points, set f′(x) = 0, then use the second derivative to classify maximum or minimum.

求驻点时，令 f′(x) = 0，然后用二阶导数判断是极大值还是极小值。

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6. Integration: Techniques and Substitution | 积分：技巧与换元

Integration is the reverse of differentiation. Know the standard integrals, including ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1.

积分是微分的逆运算。掌握标准积分公式，例如 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）。

When integrating a composite function that is linear, such as ∫ (2x+1)⁵ dx, divide by the coefficient of x: (2x+1)⁶ / 12 + C.

对线性复合函数积分时，例如 ∫ (2x+1)⁵ dx，要除以 x 的系数，结果为 (2x+1)⁶ / 12 + C。

For more complex functions, use substitution: choose u as the ‘inner’ function and remember to change dx to du.

对更复杂的函数，使用换元法：选择 u 作为“内层”函数，并记得把 dx 换成 du。

Be careful with definite integrals: change the limits when substituting, or return to the original variable before evaluating.

使用定积分时要格外小心：换元时必须改变上下限，或者先换回原变量再代入计算。

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7. Trigonometric Identities and Equations | 三角恒等式与方程

The most important identity is sin²θ + cos²θ = 1. Also remember tanθ = sinθ / cosθ and the double-angle formulas.

最重要的恒等式是 sin²θ + cos²θ = 1。还要记住 tanθ = sinθ / cosθ 以及二倍角公式。

When solving equations, find all solutions in the given interval. For example, sinθ = 0.5 has solutions at 30° and 150° in the range 0° to 180°.

解三角方程时，要找出给定区间内的所有解。例如，在 0° 到 180° 内，sinθ = 0.5 的解为 30° 和 150°。

Do not divide by a trigonometric function unless you have checked that it cannot be zero. Dividing by cosθ can lose solutions where cosθ = 0.

不要随意除以三角函数，除非你已确认它不可能等于零。除以 cosθ 可能会丢失 cosθ = 0 时的解。

Draw a quick sketch of the graph to confirm the number of solutions.

快速画出函数图像，确认解的个数。

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8. Exponentials and Logarithms | 指数与对数

Logarithms are the inverse of exponentials. The key laws are log(xy) = log x + log y and log(x/y) = log x − log y, with log xⁿ = n log x.

对数是指数的逆运算。关键法则包括 log(xy) = log x + log y、log(x/y) = log x − log y 和 log xⁿ = n log x。

A common mistake is to apply these laws incorrectly: log(x + y) ≠ log x + log y. The sum rule applies only to products inside the logarithm.

常见错误是错误使用法则：log(x + y) ≠ log x + log y。加法法则仅适用于对数内部的乘积。

To solve a × bⁿ = c, take logs of both sides first: ln(a) + n ln(b) = ln(c).

解 a × bⁿ = c 时，先两边取对数：ln(a) + n ln(b) = ln(c)。

Watch the base of the logarithm: natural log uses base e, and ‘log’ without a base is usually base 10 in non-calculator exams.

注意对数的底数：自然对数以 e 为底，而在不使用计算器的考试中，没有写底数的 log 通常以 10 为底。

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9. Coordinate Geometry and Circles | 坐标几何与圆

Know the equation of a circle: (x − a)² + (y − b)² = r². The centre is (a, b) and radius is r.

掌握圆的标准方程：(x − a)² + (y − b)² = r²，圆心为 (a, b)，半径为 r。

To find the distance from a point to a line, use the perpendicular distance formula. For a circle, the distance from the centre to a tangent line equals the radius.

求点到直线的距离，使用点到直线距离公式。对于圆，圆心到切线的距离等于半径。

A common error is missing the difference between the chord and the tangent. A tangent meets the circle at exactly one point, while a chord connects two points on the circle.

常见错误是混淆弦和切线。切线与圆只有一个交点，而弦连接圆上的两个点。

When solving the intersection of a line and a circle, substitute the line equation into the circle equation. If the discriminant is zero, the line is tangent.

求直线与圆的交点时，将直线方程代入圆方程。若判别式等于零，则直线为切线。

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10. Vectors and 3D Geometry | 向量与三维几何

Vectors describe displacement, and you should be comfortable with notation such as a = ai + bj + ck.

向量描述位移，你需要熟练使用 a = ai + bj + ck 这类表示方法。

The scalar product is a · b = |a||b| cosθ. Use it to find the angle between two vectors. If a · b = 0, the vectors are perpendicular.

标量积为 a · b = |a||b| cosθ，可用来求两

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