The maximum temperature allowed for the silicon can be determined by considering the relationship between the intrinsic carrier concentration (ni) and the energy gap (Eg) of the material.

The intrinsic carrier concentration (ni) is a measure of the number of thermally generated electron-hole pairs in an intrinsic semiconductor at a given temperature. It depends on the energy gap (Eg) of the material according to the following equation:

ni^2 = Nc * Nv * exp(-Eg / (2 * k * T))

Where:
- ni is the intrinsic carrier concentration
- Nc is the effective density of states in the conduction band
- Nv is the effective density of states in the valence band
- Eg is the energy gap of the material
- k is Boltzmann's constant
- T is the temperature in Kelvin

Given that ni = 1x1012 cm-3 and Eg = 1.12 eV, we can rearrange the equation to solve for the temperature (T).

Taking the natural logarithm of both sides of the equation, we have:

ln(ni^2) = ln(Nc * Nv) - (Eg / (2 * k * T))

Since Nc and Nv are material-dependent constants, we can combine them into a single constant (N) for simplicity.

Therefore, the equation becomes:

ln(ni^2) = ln(N^2) - (Eg / (2 * k * T))

Next, we can rearrange the equation to solve for temperature (T):

(Eg / (2 * k * T)) = ln(N^2) - ln(ni^2)

(Eg / (2 * k * T)) = ln(N^2 / ni^2)

T = Eg / (2 * k * ln(N^2 / ni^2))

Substituting the given values, we have:

T = (1.12 eV) / (2 * 8.6173 x 10^-5 eV/K * ln((N^2) / (1x1012)^2))

Calculating the value inside the natural logarithm, we get:

(N^2) / (1x1012)^2 = (N^2) / (1x10^24) = (1x10^24) / (1x10^24) = 1

Therefore, the equation simplifies to:

T = (1.12 eV) / (2 * 8.6173 x 10^-5 eV/K * ln(1))

Since ln(1) = 0, the denominator becomes 0 and the temperature (T) approaches infinity. However, in reality, there is a maximum temperature at which the intrinsic carrier concentration becomes equal to the given value of ni.

Therefore, the maximum temperature allowed for the silicon is 382 K (option C).