= Solution
The nonnegative <modulus> $r=|z|$ must be a zero of the <sine> function. The restriction $r<10$ permits $r=0,\pi,2\pi,3\pi$, since $3\pi<10<4\pi$. Thus the <locus> is \b[the origin and three concentric circles], centred at the origin with radii $\pi$, $2\pi$ and $3\pi$. A circle of radius zero is just one point.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-complex-loci.png]
{title=Complex-plane loci: sine of the modulus vanishes on three circles and the origin; real logarithm part fixes one circle}
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