= Solution
{c}
The <Pauli matrices> are <Hermitian matrices> with zero <trace>, so a real <Bloch vector> makes $\varrho$ <Hermitian> and gives $\operatorname{tr}\varrho=1$. Their <Pauli multiplication identity> implies $(\mathbf s\cdot\boldsymbol\sigma)^2=|\mathbf s|^2I$. Consequently the two <eigenvalues> of $\mathbf s\cdot\boldsymbol\sigma$ are $\pm|\mathbf s|$, including the zero-vector case, and
$$
\boxed{\lambda_\pm(\varrho)=\frac{1\pm|\mathbf s|}{2}.}
$$
A <density matrix> must also be <positive semidefinite>. The displayed <eigenvalues> are nonnegative exactly when $\boxed{|\mathbf s|\leq1}$. Thus real coordinates alone are insufficient: the allowed vectors fill the <Bloch ball>. Its boundary consists of <pure states>, with <eigenvalues> one and zero; its center is the <maximally mixed state> $I/2$.
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