= Solution
{c}
The <cardioid> has <reflection symmetry> about the $x$ axis because $1+\cos(-\theta)=1+\cos\theta$. It passes through $(2,0)$ at $\theta=0$, through $(0,\pm1)$ at $\theta=\pm\pi/2$, and has a <cusp> at the origin when $\theta=\pi$. These points and the following sketch show its orientation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii-cardioid.png]
{title=The cardioid and unit circle, with the three regions between them shaded}
Using the <polar coordinates> formula for <arc length>,
$$
L=\int_0^{2\pi}\sqrt{(1+\cos\theta)^2+\sin^2\theta}\,d\theta
=\int_0^{2\pi}2\left|\cos\frac\theta2\right|\,d\theta.
$$
The <absolute value> changes sign at $\theta=\pi$, so
$$
\boxed{L=4\int_0^\pi\cos\frac\theta2\,d\theta=8.}
$$
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