2025 · Paper 1 · Mathematics
Question 9
The printed question, the official key, and each model’s JSON answer plus the reasoning it wrote.
Q09
Official 1.15 – 1.25
Model answers
Claude Fable 5.1
Anthropic
1.2
Claude Fable 5.1 reasoning
Let position vectors be p, q, r for P, Q, R and s for S. From SP + 5 SQ + 6 SR = 0: (p − s) + 5(q − s) + 6(r − s) = 0 ⟹ s = (p + 5q + 6r)/12. Midpoints: E = (p + r)/2, F = (q + r)/2. EF = F − E = (q − p)/2. ES = S − E = (p + 5q + 6r)/12 − (6p + 6r)/12 = (5q − 5p)/12 = (5/12)(q − p). Both vectors are parallel to (q − p), so |EF| / |ES| = (1/2) / (5/12) = 12/10 = 1.2. So S lies on segment EF, and the required ratio is 6/5 = 1.2.
GPT 6 Astra
OpenAI
1.2
GPT 6 Astra reasoning
No written reasoning was returned for this item.
GPT 5.6 Luna
OpenAI
1.2
GPT 5.6 Luna reasoning
No written reasoning was returned for this item.