= Solution
Use a cold, unmagnetized, collisionless <plasma> with a sharp interface, and take the incident medium to be vacuum. Its <relative permittivity> is $\epsilon_r=1-\omega_p^2/\omega^2$, so at the stated <frequencies> $n=\sqrt{1-(10/12)^2}=\sqrt{11}/6$. Conservation of tangential <wavenumber> gives
$$
k_{z,\rm plasma}^2=\frac{\omega^2}{c^2}\left(\cos^2\theta_i-\frac{\omega_p^2}{\omega^2}\right).
$$
For <oblique total reflection at an unmagnetized plasma>, require $k_{z,\rm plasma}^2<0$, or $\cos\theta_i<5/6$. Thus \b[total reflection occurs for $\theta_i>33.56^\circ$ measured from the normal], up to grazing incidence. At the ideal critical <angle> $\theta_c=\cos^{-1}(5/6)$ the transmitted normal <energy> flux also vanishes, so the lossless reflectance reaches unity there. Collisions or an imposed <magnetic field> change this simple criterion. The constitutive response used here follows the electron-motion derivation in the cited university notes.<Oscillations and Waves by Richard Fitzpatrick>[Section 9.7, equations 9.27–9.30, unmagnetized-plasma dispersion and refractive index]{ref}
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