AS Mathematics: High-Frequency Exam Topics Summary | AS 数学：高频考点总结

📚 AS Mathematics: High-Frequency Exam Topics Summary | AS 数学：高频考点总结

Getting ready for your AS Mathematics exam? Whether you are taking Pure Mathematics 1 alone or alongside Statistics or Mechanics, certain topics consistently appear in exams. This summary revisits the high-frequency concepts, formulas, and typical question types to help you focus your revision.

准备AS数学考试？不管你是考纯数1还是搭配统计或力学，某些主题总是高频出现。本总结回顾这些常考概念、公式和典型题型，助你集中复习。

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1. Quadratic Equations and Inequalities | 二次方程与不等式

Quadratic equations in the form ax² + bx + c = 0 are solved by factorising, completing the square, or using the quadratic formula x = [−b ± √(b² − 4ac)] / (2a). Always check whether the question requires exact form or decimal answers.

形如ax² + bx + c = 0的二次方程可通过因式分解、配方法或求根公式 x = [−b ± √(b² − 4ac)] / (2a) 求解。务必看清题目要求精确值还是小数答案。

The discriminant Δ = b² − 4ac tells you about the roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots. You may be asked to find the range of k for which a quadratic has real roots.

判别式Δ = b² − 4ac给出根的信息：Δ > 0时有两个不等实根；Δ = 0时有一个重根；Δ < 0时无实根。常考求k的范围使二次方程有实根。

To solve a quadratic inequality, e.g. x² − 5x + 6 > 0, first find the roots and then sketch the parabola to identify the intervals where the inequality holds. Use critical values and test intervals.

解二次不等式，如x² − 5x + 6 > 0，先求根，然后画抛物线草图来确定满足不等式的区间。使用关键值并检查区间。

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2. Functions and Graph Transformations | 函数与图像变换

For a function f(x), the domain is the set of all possible input values (x), and the range is the set of all possible output values y. Watch out for restrictions like square roots (argument ≥ 0) and denominators (≠ 0).

函数f(x)的定义域是所有可能输入x的集合，值域是所有可能输出y的集合。注意限制条件，如平方根内≥0，分母≠0。

Composite functions such as fg(x) mean applying g then f. For inverse function f⁻¹(x), swap x and y, then solve for y. Only one-to-one functions have inverses over their whole domain.

复合函数fg(x)意为先g后f。求反函数f⁻¹(x)时，交换x和y再解出y。只有一一映射函数在整个定义域上才有反函数。

Remember the common transformations: y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by −a; y = a f(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a. Reflections include y = −f(x) and y = f(−x).

记住常见变换：y = f(x) + a 上下平移a；y = f(x + a) 左右平移−a；y = a f(x) 垂直拉伸a倍；y = f(ax) 水平拉伸1/a倍。反射变换有y = −f(x)和y = f(−x)。

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3. Coordinate Geometry of Straight Lines | 直线坐标几何

The gradient m between (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). Equation forms: y = mx + c (slope-intercept), y − y₁ = m(x − x₁) (point-slope), and ax + by + c = 0 (general form).

两点间梯度m = (y₂ − y₁)/(x₂ − x₁)。直线方程形式：y = mx + c（斜截式），y − y₁ = m(x − x₁)（点斜式），以及ax + by + c = 0（一般式）。

Parallel lines have equal gradients (m₁ = m₂). Perpendicular lines satisfy m₁ × m₂ = −1. Always convert to the form y = mx + c to compare gradients.

平行线梯度相等(m₁ = m₂)。垂直线满足m₁ × m₂ = −1。总是化为y = mx + c形式来比较梯度。

The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The distance between two points is √[(x₂ − x₁)² + (y₂ − y₁)²]. These are often used in circle geometry and coordinate proofs.

中点为((x₁ + x₂)/2, (y₁ + y₂)/2)。两点间距离为√[(x₂ − x₁)² + (y₂ − y₁)²]。这些常用于圆的几何和坐标证明。

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4. Circles | 圆的方程

The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². Expanding gives x² + y² + 2gx + 2fy + c = 0, where centre is (−g, −f) and radius = √(g² + f² − c).

圆心(a,b)、半径r的圆的方程为(x − a)² + (y − b)² = r²。展开得x² + y² + 2gx + 2fy + c = 0，圆心为(−g, −f)，半径= √(g² + f² − c)。

A tangent to a circle touches it at exactly one point. The tangent is perpendicular to the radius at the point of contact. To find intersection of a line and circle, substitute the line equation into the circle’s equation and solve; discriminant determines tangency (Δ = 0).

圆的切线只接触圆于一点。切线在切点处垂直于半径。求直线与圆的交点，将直线方程代入圆方程求解；判别式Δ=0时相切。

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5. Sequences: Arithmetic and Geometric | 数列：等差与等比

An AP has first term a, common difference d. nth term: uₙ = a + (n − 1)d. Sum of first n terms: Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l) where l is the last term. Recognise word problems involving arithmetic sequences.

等差数列首项为a，公差为d。第n项: uₙ = a + (n − 1)d。前n项和: Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，l为末项。识别涉及等差数列的应用题。

GP: first term a, common ratio r. nth term: uₙ = a rⁿ⁻¹. Sum of first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Sum to infinity (convergent GP when |r| < 1): S∞ = a/(1 − r). Common exam questions ask for the least n for Sₙ to exceed a value.

等比数列首项a，公比r。第n项: uₙ = a rⁿ⁻¹。前n项和: Sₙ = a(1 − rⁿ)/(1 − r) (r≠1)。无穷和（|r|<1时收敛）: S∞ = a/(1 − r)。常见考题要求求最小n使得Sₙ超过某值。

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6. Trigonometry | 三角函数

sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. Key identities: sin² θ + cos² θ = 1, tan θ = sin θ / cos θ. Be comfortable with both degrees and radians.

sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边。关键恒等式：sin² θ + cos² θ = 1, tan θ = sin θ / cos θ。熟悉度和弧度制。

Solve equations like sin x = 0.5 for 0° ≤ x ≤ 360° or 0 ≤ x ≤ 2π. Use CAST diagram or graph to find all solutions. Remember to adjust the interval if the argument is 2x or (x + 30°).

解方程如sin x = 0.5，在0°≤x≤360°或0≤x≤2π内。用CAST图或图像求所有解。若参数为2x或(x+30°)，记得调整区间。

Know the shapes of y = sin x, y = cos x, y = tan x. Amplitude and period changes: y = a sin(bx) has amplitude |a| and period 2π/b. Phase shifts and vertical shifts may be tested.

掌握y = sin x, y = cos x, y = tan x的图像。振幅和周期变化：y = a sin(bx)振幅为|a|，周期为2π/b。可考相移和垂直平移。

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7. Differentiation Basics | 微分基础

If y = xⁿ, dy/dx = n xⁿ⁻¹. For sums, differentiate term by term. Constants differentiate to zero. The derivative of a constant times a function is constant times the derivative.

若y = xⁿ，dy/dx = n xⁿ⁻¹。对求和逐项求导。常数

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