= Solution
The <Bloch vectors> of $|a\rangle$ and $|b\rangle$ are
$$
\mathbf r_a=(1/\sqrt2,0,1/\sqrt2),\qquad
\mathbf r_b=(1/\sqrt2,0,-1/\sqrt2).
$$
They share the same $x$ component and differ along $z$. The <equal-prior discrimination of qubit states> therefore selects the $Z$ axis: it maximizes the separation of their projected outcome probabilities. This also explains why the $X$ measurement in part 3 is uninformative.
For $|0\rangle$ and $|+\rangle$, the Bloch vectors are $(0,0,1)$ and $(1,0,0)$. Their difference points along $\mathbf n=(-1,0,1)/\sqrt2$. Hence <Carl W. Helstrom>'s <Helstrom measurement for the zero and plus states> measures
$$
\boxed{M_{\rm opt}=\frac{Z-X}{\sqrt2},\qquad
P_\pm=\frac{I\pm M_{\rm opt}}2.}
$$
Guess $|0\rangle$ after $P_+$ and $|+\rangle$ after $P_-$. An explicit <orthonormal basis> for these projectors is
$$
|h_+\rangle=\cos(\pi/8)|0\rangle-\sin(\pi/8)|1\rangle,\qquad
|h_-\rangle=\sin(\pi/8)|0\rangle+\cos(\pi/8)|1\rangle.
$$
The positive measurement axis is tilted by $\pi/4$ from $+z$ towards $-x$ on the <Bloch sphere>; the corresponding state-vector rotation angle is $-\pi/8$.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2008/5-helstrom-bloch-geometry.png]
{title=Bloch-sphere cross-sections for the two equal-prior discrimination problems}
{description=Solid arrows are the candidate-state <Bloch vectors>. Dashed lines show the optimal measurement axes. The first pair has identical $x$ projections; the second is separated most strongly along $(z-x)/\sqrt2$.}
For completeness, if a projective measurement guesses $0$ along a unit Bloch axis $\mathbf n$, its average success is $1/2+(n_z-n_x)/4$. The <Cauchy-Schwarz inequality> bounds $n_z-n_x\le\sqrt2$, with equality at the axis above. The <Holevo–Helstrom theorem> shows this optimum also holds among general <POVMs>. Thus
$$
\boxed{P_{\rm success}^{\rm opt}=\frac12\left(1+\frac1{\sqrt2}\right),\qquad
P_{\rm error}^{\rm opt}=\frac12\left(1-\frac1{\sqrt2}\right).}
$$
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