During the whole question it is really confusing which a is being used, it would help a lot if in the answers they would use an a0 and an a_new term or something like that because in the answers they say ka = pi, but this is a_new instead of a0 and they don’t mention switching between them even though they do.
Yes, this is a fair point: the notation is easy to mix up here.
In the diatomic-chain formula, a is the length of the diatomic unit cell. That unit cell contains two atoms. If the nearest-neighbour spacing of the corresponding monatomic chain is called a_0, then
a_{\mathrm{cell}} = 2a_0.
So when the solution says that the gap opens at
ka = \pi,
the a there means a_{\mathrm{cell}}, not the nearest-neighbour spacing a_0. Written with both symbols, this is
k a_{\mathrm{cell}} = \pi
\quad \Leftrightarrow \quad
k = \frac{\pi}{2a_0}.
This is the reason why the monatomic-chain dispersion gets folded back if we artificially choose a unit cell that is twice as large. When m_1=m_2, this folding does not open a gap. When m_1 and m_2 become
slightly different, the doubled unit cell becomes physically meaningful, and a gap opens at the edge of the smaller Brillouin zone.
So your interpretation is correct: the solution is using the doubled unit-cell length in ka=\pi. I’ll clarify this right away.