Step 1: Judge statement I.
Boron is a hard solid bound by a strong covalent network, which gives it a very high melting point. So statement I is correct.
Step 2: Judge statement II.
Boron is actually a low density element and is not known for high density, so statement II is incorrect.
Step 3: Judge statement III.
Pure boron behaves like a semiconductor, not a good metallic conductor, so its electrical conductivity is low at ordinary temperature. Statement III is incorrect.
Step 4: Judge statement IV.
The $^{10}B$ isotope absorbs neutrons strongly, which is why boron is used in nuclear reactor control rods. So statement IV is correct.
Step 5: Collect the correct statements.
Only I and IV are correct, while II and III are wrong.
Step 6: Choose the option.
The right combination is I and IV only, which is option (4). \[ \boxed{\text{I, IV only}} \]