= Solution
For the two free <real scalar fields>, let $\mathcal L=\mathcal L_1+\mathcal L_2$. The infinitesimal <internal rotation symmetry of two real scalar fields> has $\delta\phi_1=\theta\phi_2$ and $\delta\phi_2=-\theta\phi_1$, with constant $\theta$. Its kinetic terms are invariant because this is an orthogonal rotation of the two-component field.
Under the assumed invariance of the full <Lagrangian density>, <Noether's theorem> gives the current per unit transformation parameter:
$$
j^\mu=\sum_{a=1}^2\frac{\partial\mathcal L}{\partial(\partial_\mu\phi_a)}\frac{\delta\phi_a}{\theta}
=\phi_2\partial^\mu\phi_1-\phi_1\partial^\mu\phi_2.
$$
The <Noether current> and <Noether charge> are therefore
$$
\boxed{j^\mu=\phi_2\partial^\mu\phi_1-\phi_1\partial^\mu\phi_2,\qquad
Q=\int d^3x\,(\phi_2\pi_1-\phi_1\pi_2).}
$$
Independent species commute at equal time, so these products require no extra ordering sign. With vanishing boundary flux, $Q$ is conserved. Part (vii) checks precisely what the symmetry assumption requires of the masses.
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