Compulsory Part (Mathematics): Core Rules, Worked Example and Traps
What the Compulsory Part tests
The Compulsory Part is the common core of HKDSE Mathematics, assessed in Paper 1 (conventional questions, 65%, 2 h 15 min) and Paper 2 (multiple choice, 35%, 1 h 15 min). It is organised into three strands: Number and Algebra, Measures, Shape and Space, and Data Handling. Questions are cumulative: a quadratic-model question can require completing the square and interpreting the vertex; a 3D trigonometry question can combine the sine formula, the cosine formula, and Pythagoras' theorem. Paper 1 rewards clear method marks; Paper 2 rewards exact answers and quick elimination.
Core rules to have automatic
- Quadratic equations: discriminant Δ = b² − 4ac; Δ > 0 two real roots, Δ = 0 one repeated root, Δ < 0 no real roots. Completing the square gives vertex (−b/(2a), c − b²/(4a)) for y = ax² + bx + c.
- Exponential/log: log(ab) = log a + log b, log(a/b) = log a − log b, log aⁿ = n log a; changing base log_b x = log x / log b. For a>0, a≠1, a^x = a^y ⇒ x = y.
- Sequences: arithmetic T(n) = a + (n−1)d, S(n) = n/2[2a + (n−1)d]; geometric T(n) = ar^{n−1}, S(n) = a(rⁿ − 1)/(r − 1) for r ≠ 1. For geometric sum to infinity, |r| < 1 and S∞ = a/(1 − r).
- Trigonometry: sine formula a/sin A = b/sin B = c/sin C; cosine formula a² = b² + c² − 2bc cos A; area = ½ab sin C. Heron's formula: area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2.
- Statistics: standard deviation from raw data σ = √[Σ(x−x̄)²/n]; adding a constant shifts mean but not σ; multiplying by k multiplies σ by |k|.
Worked example
A geometric sequence has second term 6 and fifth term 48. Find the first term and the sum of the first 10 terms.
Let first term a and common ratio r. ar = 6 and ar⁴ = 48. Dividing: r³ = 8, so r = 2. Then a = 6/2 = 3. S₁₀ = 3(2¹⁰ − 1)/(2 − 1) = 3(1024 − 1) = 3069.
Traps that cost marks
- Forgetting to take the square root after averaging the squared deviations; standard deviation is in the same units as the data.
- In the cosine formula, using the side opposite the angle on the left; mislabelling which angle is included.
- For log equations, checking the domain: log x requires x > 0; rejecting extraneous roots.
- For geometric series, applying S∞ only when |r| < 1; otherwise the series diverges.
- In Paper 1, not showing the substitution into the formula; method marks are awarded for the shown step, not just the final value.