2025 · Paper 2 · Mathematics
Question 13
The printed question, the official key, and each model’s JSON answer plus the reasoning it wrote.
MathematicsPaper 2Numerical
Q13
Official -2
Model answers
Claude Fable 5.1
Anthropic
Correct4 / 4
-2
Claude Fable 5.1 reasoning
ω = e^{2πi/3}. -ω = e^{iπ}e^{2πi/3} = e^{i5π/3} = e^{-iπ/3}. Let ζ = -ω = e^{-iπ/3}, a primitive 6th root of unity. Sum_{n=1}^{2025} ζ^n. Since ζ^6 = 1 and sum over full period is 0, 2025 = 6·337 + 3, so sum = ζ + ζ^2 + ζ^3 = e^{-iπ/3} + e^{-2iπ/3} + e^{-iπ} = (1/2 - i√3/2) + (-1/2 - i√3/2) + (-1) = -1 - i√3. arg(-1 - i√3) = -2π/3. So α = -2π/3, 3α/π = -2.
GPT 6 Astra
OpenAI
Correct4 / 4
-2
GPT 6 Astra reasoning
No written reasoning was returned for this item.
GPT 5.6 Luna
OpenAI
Correct4 / 4
-2
GPT 5.6 Luna reasoning
No written reasoning was returned for this item.