Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 125 / 5 / b / i / Solution

For $P=(1,5)$, the tangent slope issoFor $Q=(-1,1)$, the secant slope is $2$, giving

Let $T=(0,0)$, the rational point of order two. Direct use of the chord-and-tangent law givesTranslation by a torsion point sends torsion points to torsion points, proving the claim.

Solved by gpt-5.6-sol high.