= Solution
With mixing and $\gamma=|\beta_{sp}|$, the equal-on-site-energy result is
$$
\boxed{E_\pm(k)=\pm2\sqrt{t^2\cos^2(ka)+\gamma^2\sin^2(ka)}.}
$$
For the regime $0<\gamma<t$ considered in part (iii), the crossing becomes an <avoided crossing> with gap $4\gamma$ at $k=\pi/(2a)$. The lower band has its maximum there and the upper band its minimum. Their orbital characters exchange smoothly across this region, with equal $s/p$ weights at its centre. The lower-right panel gives the sketch.
The zero-mixing limit recovers the crossing of case (1). A useful special limiting case is $\gamma=t$, where the square root is independent of <wavevector> and both bands are flat at $E=\pm2t$. This flat-band limit is outside the strict $\gamma<t$ assumption made in part (iii).
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