Mathematics 压轴题: Final Problem Mastery
The Gaokao Mathematics paper (满分150分) contains 22 questions across multiple choice (12 × 5 points), fill-in-the-blank (4 × 5 points), and extended response (6 questions, 70 points total). The final two extended response questions (Q21 and Q22) are the 压轴题 — the highest-difficulty questions, worth 12 marks each and specifically designed to separate students who score in the 135-150 range from those who score 120-134.
Q21 is typically a complex function or inequality problem requiring multiple derivative applications, often with a parameter. Common structure: Part (1) is straightforward (tangent line, monotonic interval) — earning 4 marks; Part (2) introduces a parameter (if the function f(x, a) has exactly two critical points in a given interval, find the range of a) — earning 4 marks; Part (3) asks to prove or find conditions for a more complex relationship using Parts 1-2 as groundwork — earning 4 marks. Students targeting 700+ must complete Parts 1-2 of Q21 reliably and make meaningful progress on Part 3 — a correct setup of Part 3 with an incomplete algebraic finish typically earns 2-3 partial marks.
Q22 is typically a sequence (数列) problem, complex geometry (解析几何 — conic sections), or a linear algebra application. Conic section problems (椭圆/抛物线/双曲线) are common: given the standard equation of a curve and conditions about a line passing through a specific point, find the locus of a related point or prove a specific geometric property. These require fluency with: standard conic equations, the relationship between discriminant and number of intersections, parametric representation of points on a conic, and algebraic manipulation of high-degree expressions.
For 700+ targeting, the strategy for 压轴题 is not to spend all remaining time on Part 3 and leave other questions incomplete. Optimal time management: complete all of Q1-Q18 first, then Q19-Q20, then return to Q21 Parts 1-2 and Q22 Parts 1-2, then — if time permits — attempt Q21 Part 3 and Q22 Part 3. This strategy guarantees 130+ (Parts 1-2 fully completed) and gives a realistic chance at 140+ (meaningful progress on Parts 3). Attempting Part 3 first and running out of time for earlier questions caps the score at 120-125.