= Solution
{c}
For the given <wrapped hyperbolic cosine curve>,
$$
x^2+y^2=\cos^2t+\sin^2t=1,\qquad z=\cosh t\geq1>0,
$$
so $\boxed{C\subset S}$. The <curve> makes one full turn around the <circular cylinder>, descending from $(-1,0,\cosh\pi)$ to $(1,0,1)$ and ascending to its initial point. Its two parameter endpoints coincide; their <tangent vectors> $(0,-1,-\sinh\pi)$ and $(0,-1,\sinh\pi)$ do not agree, so the resulting closed <curve> has a corner there.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v-cylinder-curve.png]
{title=The <hyperbolic cosine> <curve> wrapped around a unit <circular cylinder>, and the same <surface> unrolled}
{description=The shaded finite patch extends from $z=0$ to $z=\cosh t$. On the unrolled <cylinder>, its horizontal coordinate is $t\in[-\pi,\pi]$; the two vertical edges are identified.}
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