Which expression is the most appropriate for intrinsic carrier concentration?

The standard intrinsic concentration depends on the geometric mean of the effective density of states and an activation term with $E_g/2$.

Correct answer: $n_i = \sqrt{N_C N_V}\, e^{-E_g/(2k_B T)}$

What physical condition justifies the Maxwell-Boltzmann approximation for carriers?

The approximation holds when $f(E) \ll 1$ near the states of interest.

Correct answer: That the relevant band-edge is many $k_B T$ away from the Fermi level so the occupation is low and nondegenerate.

In an n-type semiconductor at moderate temperature, the Fermi level typically shifts:

Donor doping increases electron concentration and pushes $E_F$ toward $E_C$.

Correct answer: Toward the conduction band

Why can the intrinsic Fermi level deviate from the exact midgap position?

The intrinsic level depends on $N_C$ and $N_V$, not only on $E_g$.

Correct answer: Because the conduction and valence bands usually have different effective densities of states, so asymmetry shifts the equilibrium reference level.