In a semiconductor, there are:(A) no free electrons at any temperatur…

The correct answer is Option (2) → (B) and (C) only

In a semiconductor the energy gap between the valence band and the conduction band is small (about 1 eV). Electrons become free only when thermal energy lifts them across this gap into the conduction band.

(A) no free electrons at any temperature — Incorrect. At room temperature some electrons gain enough thermal energy to cross the gap, so free electrons are present and the semiconductor conducts a little.

(B) no free electrons at 0 K — Correct. At 0 K there is no thermal energy: the valence band is full and the conduction band is empty, so the semiconductor behaves as an insulator.

(C) number of free electrons increase with temperature — Correct. A higher temperature lifts more electrons across the gap, so the number of free electrons (and of holes) rises and the resistivity falls.

(D) number of free electrons decrease with temperature — Incorrect. This is the opposite of (C); heating adds free electrons, it does not remove them.

NCERT (2026-27):
Intrinsic Semiconductor – “… An intrinsic semiconductor will behave like an insulator at T = 0 K as shown in Fig. 14.6(a). It is the thermal energy at higher temperatures (T > 0K), which excites some electrons from the valence band to the conduction band.”