The gaokao mathematics paper is worth 150 points and usually lasts two hours. On the national papers used by most provinces in recent years it has consisted of single-answer multiple choice, multiple-answer multiple choice with partial credit, fill-in-the-blank questions, and five or six long-form solution questions. The exact number of questions has changed between recent years, and some provinces set their own paper.
The topics will be familiar to anyone who has studied upper-secondary maths elsewhere: functions and derivatives, trigonometry and solving triangles, sequences, solid geometry (often using vectors), probability and statistics, and conic sections. What makes the paper hard is not exotic content. It is the combination of no calculator, tight timing and a few final questions designed to separate the strongest candidates.
Here is a question of the kind that appears in the middle of the solution section. It is written for this article, not taken from a past paper. A sequence has a₁ = 1 and aₙ₊₁ = 2aₙ + 1. (1) Show that {aₙ + 1} is a geometric sequence and find aₙ. (2) Find the sum of the first n terms, Sₙ.
Part (1): add 1 to both sides of the recurrence to get aₙ₊₁ + 1 = 2aₙ + 2 = 2(aₙ + 1). Since a₁ + 1 = 2, which is not zero, {aₙ + 1} is geometric with first term 2 and common ratio 2. So aₙ + 1 = 2ⁿ, and aₙ = 2ⁿ − 1.
Part (2): Sₙ = (2 + 4 + … + 2ⁿ) − n. The geometric part sums to 2ⁿ⁺¹ − 2, so Sₙ = 2ⁿ⁺¹ − n − 2. A quick check catches slips: S₁ = 4 − 1 − 2 = 1 and S₂ = 8 − 2 − 2 = 4, which match a₁ = 1 and a₁ + a₂ = 1 + 3.
Notice what is being tested: recognising a standard transformation, stating why the sequence is geometric rather than just asserting it, and clean algebra. Marking is step-based (步骤分), so a candidate who writes the justification and the correct general term earns marks even if the sum goes wrong. Show the reasoning, not just the answer.
The difficulty comes at the end. The final one or two questions (压轴题) typically combine derivatives with inequalities, or conic sections with lengthy algebra, or introduce an unfamiliar definition the candidate has to work with on the spot. Many well-prepared students aim to secure every point before these questions and treat the last parts as a bonus. For an international learner, that is also the right strategy: master the middle of the paper first.