Mathematics Compulsory | DSE Core Math for Business

Paper 1 Conventional Questions (65%, 2h15m) | Paper 2 Multiple Choice (35%, 1h15m, 45 questions)

Chapter 1 Number & Algebra ★★★

Must-memorise Identities

(a+b)² = a² + 2ab + b²        (a−b)² = a² − 2ab + b²
a² − b² = (a+b)(a−b)
a³ + b³ = (a+b)(a² − ab + b²)  a³ − b³ = (a−b)(a² + ab + b²)
Indices:  aᵐ·aⁿ = aᵐ⁺ⁿ   (aᵐ)ⁿ = aᵐⁿ   a⁻ⁿ = 1/aⁿ
Logs:  log(MN) = logM + logN   log(M/N) = logM − logN   log Mᵏ = k·logM

Estimation & Errors

• Maximum absolute error = ½ × scale interval
• Relative error = absolute error ÷ measured value
• Percentage error = relative error × 100%

Chapter 2 Functions & Graphs ★★★

Quadratic Functions（每年必考）

For y = ax² + bx + c:

• Vertex form: y = a(x−h)² + k, vertex (h, k)
• Axis of symmetry: x = −b / 2a
• Discriminant Δ = b² − 4ac：Δ>0 two distinct real roots；Δ=0 one double root；Δ<0 no real roots
• Sum of roots = −b/a；Product of roots = c/a

Graph Transformations

Transformation	Effect on y = f(x)
y = f(x) + k	translate upward by k
y = f(x − h)	translate rightward by h
y = k·f(x)	vertical stretch (×k)
y = f(kx)	horizontal shrink to 1/k
y = −f(x) / f(−x)	reflect about x-axis / y-axis

Chapter 3 Equations & Inequalities ★★

• Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
• Quadratic inequality: sketch the parabola first, then read the solution range（先画图后写解）
• Linear programming: constraints → feasible region → test vertices for optimal value（商科应用高频：利润最大化）

Chapter 4 Sequences ★★★（复利/投资高频载体）

	Arithmetic (A.S.)	Geometric (G.S.)
nth term	a + (n−1)d	a·rⁿ⁻¹
Sum of n terms	(n/2)[2a + (n−1)d]	a(1 − rⁿ) / (1 − r)
Sum to infinity	—	a / (1 − r)，only when |r| < 1

Business application：分期存款、按揭还款、机器折旧均为 G.S. 模型。

Chapter 5 Probability & Statistics ★★★（商科最高权重）

Probability

P(A∪B) = P(A) + P(B) − P(A∩B)
Independent:   P(A∩B) = P(A) × P(B)
Conditional:   P(A|B) = P(A∩B) / P(B)
Permutation:   nPr = n! / (n−r)!
Combination:   nCr = n! / [r!(n−r)!]

• 抽签不放回 → 用组合数或树状图，切勿当成独立事件

Statistics

Measure	Key points
Mean / Median / Mode	Mean 受极端值影响大；skewed data 用 median
Standard deviation σ	加常数 k → σ 不变；乘 k → σ 变 σ×|k|
Inter-quartile range	IQR = Q₃ − Q₁，box-and-whisker plot 必考
Standard score	z = (x − mean) / σ（跨组比较表现）
Normal distribution	68% within ±1σ，95% within ±2σ，99.7% within ±3σ

Sampling 陷阱：questionnaire at MTR station ≠ random sample（每年 MC 必考抽样偏差判断）。

Chapter 6 Coordinate Geometry & Trigonometry ★★

Distance = √[(x₁−x₂)² + (y₁−y₂)²]        Mid-point = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope m = (y₁−y₂)/(x₁−x₂)                Perpendicular: m₁ × m₂ = −1
Circle x² + y² + Dx + Ey + F = 0:  centre (−D/2, −E/2), radius √(D²/4 + E²/4 − F)
Sine rule:  a/sinA = b/sinB = c/sinC
Cosine rule:  c² = a² + b² − 2ab·cosC
Area of triangle = ½·ab·sinC

Business-oriented Special Module（弥补无 M1/M2 短板）★★★

1. Compound Interest & Growth

Compound amount:   A = P(1 + r/n)^(nt)
Depreciation:      A = P(1 − r)^t
Real value = nominal value ÷ (1 + inflation rate)^t

Worked example：Deposit $50,000 at 4% p.a. compounded quarterly for 3 years: A = 50000 × (1 + 0.04/4)^12 = 50000 × (1.01)^12 ≈ $56,341

2. Percentage Change Chain（对接 Economics 弹性计算）

• Overall change = (1 + a%)(1 + b%) − 1，不可直接相加
• GDP growth rate、CPI 通胀率均为此模型

3. Statistical Graph Interpretation in English

Standard sentences for describing graphs:

• The bar chart shows that X accounted for the largest share (35%) of...
• There was a steady upward trend in ... from 2020 to 2024, rising by 12%.
• The median of Group A is higher than that of Group B, while Group A also shows a larger spread (IQR).

English Writing Standard for Long Questions

1. Every step must state the reason: By the cosine rule, ... / By similar triangles (AAA), ...
2. Conclusion sentence is compulsory: ∴ The claim is agreed/disagreed because...
3. Keep intermediate values in 4+ significant figures; round only the final answer to 3 sig. fig. unless specified
4. Write units in every final answer ($, cm², km/h)

Careless Error Checklist（考前默读）

• [ ] 负号：移项、展开括号后逐项检查
• [ ] 单位：cm ↔ m、% ↔ 小数换算
• [ ] MC 题：算完后代回验算 / 用排除法双保险
• [ ] Probability 答案是否在 0–1 之间；总和是否合理
• [ ] 图形题：答案与图的直观大小是否吻合

Unit Test Prompt

You are an HKDSE Mathematics (Compulsory Part) marker. For the chapter I specify, design 6 questions: 3 MC + 2 short conventional + 1 long application question with business/economic context (compound interest, statistics or linear programming).
After I answer, check every calculation step, point out omitted steps that would lose marks under HKEAA marking, and identify any careless-error patterns.

Chapter: [e.g. Probability & Statistics]