= Solution
Define $s_i=\sin[2\pi(t_i-T)/P+\pi]$ and use the independent quoted Gaussian errors $\sigma=10\,\mathrm{m\,s^{-1}}$, with period and <exoplanet transit> epoch fixed. The model is linear in $K,V_0$ despite its sinusoidal <time> dependence. Minimize $\chi^2=\sum_i(V_i-Ks_i-V_0)^2/\sigma^2$. With $N=5$ the <normal equations> are
$$
\begin{pmatrix}\sum s_i^2&\sum s_i\\\sum s_i&N\end{pmatrix}
\begin{pmatrix}K\\V_0\end{pmatrix}
=\begin{pmatrix}\sum s_iV_i\\\sum V_i\end{pmatrix}.
$$
Using $\sum s_i=-0.741472$ and $\sum s_i^2=2.824633$, the solution is
$$
\boxed{K=59.26\,\mathrm{m\,s^{-1}},\qquad
V_0=-15348.41\,\mathrm{m\,s^{-1}}.}
$$
The residuals in observation order are $(3.67,11.01,2.74,-7.90,-9.51)\,\mathrm{m\,s^{-1}}$. Therefore $\chi^2_{\min}=2.950$ for $5-2=3$ <degrees of freedom>, giving reduced $\chi^2=0.983$. The upper-tail <probability> is about $0.40$: \b[the fit is consistent with the stated errors], although five points do not establish a unique physical explanation.
For known errors the <covariance matrix> is $\sigma^2(X^TX)^{-1}$. Equivalently, profiling out the fitted offset gives
$$
\sigma_K^2=\frac{\sigma^2}{\sum_i(s_i-\bar s)^2},\qquad
\boxed{\sigma_K=6.07\,\mathrm{m\,s^{-1}}.}
$$
This is the $\Delta\chi^2=1$ one-parameter uncertainty, not the curvature with $V_0$ incorrectly held fixed. Propagating it through the supplied <mass> relation, and treating its coefficient and $P$ as fixed,
$$
\boxed{m_p=(0.347\pm0.035)\,M_J.}
$$
The symbol $\pm nP$ in the <exoplanet transit> epoch denotes periodic copies of the same <exoplanet transit> phase, not an uncertainty <distribution> to propagate. No independent error for the empirical <mass> coefficient was supplied.
Additional independent observations retain positive weight even with twice the error: each has one quarter of the previous per-observation weight. Their information adds to the existing information <matrix>, so the formal <mass> uncertainty \b[decreases for informative additional orbital phases and cannot increase] under this fixed-error model. It can remain unchanged in a degenerate phase configuration that supplies no further <amplitude> information after fitting the offset. The conclusion assumes the same correct model and independent errors, not a new fitted error rescaling or unmodelled systematic effects.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2014/paper-1-radial-velocity.png]
{title=Original plot of the fitted orbital radial-velocity variation and residuals; observed numbers used without reproducing the source table}
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