= Solution
The opposite spatial parity of the <s orbital> and axial <p orbital> gives opposite signs for left and right $s$–$p$ hopping. With real $\beta_{sp}$, one consistent Bloch-basis convention gives
$$
H(k)=\begin{pmatrix}
E_s(k)&2i\beta_{sp}\sin(ka)\\
-2i\beta_{sp}\sin(ka)&E_p(k)
\end{pmatrix}.
$$
The resulting energies are
$$
\boxed{E_\pm=\frac{E_s+E_p}{2}
\pm\sqrt{\left(\frac{E_s-E_p}{2}\right)^2+4\beta_{sp}^2\sin^2(ka)}.}
$$
Thus <s-p mixing in a one-dimensional tight-binding chain> lowers the lower branch and raises the upper branch wherever the coupling is nonzero. For separated atomic levels, the lower branch remains mainly $s$ and the upper mainly $p$, with increasing admixture where their uncoupled energies approach.
The upper-right panel of the figure in part (a)(ii) compares the mixed curves with dashed unmixed curves. The mixing vanishes at both $k=0$ and $\pi/a$, so those endpoints remain unchanged. If the unmixed curves approach or cross in the interior, the nonzero coupling produces an <avoided crossing>.
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