= Solution
Let the common on-site <energy> be zero and the common hopping magnitude be $t=|\beta|$. Consistent with the axial-$p$ convention and the formula in part (iii), the hopping signs are $\beta_s=-t$ and $\beta_p=+t$.
Without mixing,
$$
\boxed{E_s(k)=-2t\cos(ka),\qquad E_p(k)=+2t\cos(ka).}
$$
The curves cross at \b[$k=\pi/(2a)$, $E=0$], with no gap. The lower-left panel shows the continuous pure-$s$ and pure-$p$ branches. Their identities remain fixed through the crossing; the energy-ordered lower state changes from $s$ to $p$ character on passing through it.
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