Step 1: Analyse Statement--I.
Lead storage battery is a rechargeable battery.
Hence, it is a secondary battery. \checkmark
Step 2: Analyse Statement--II.
The positive plate contains \(\mathrm{PbO_2}\) and the negative plate contains Pb. However, the statement regarding cathode and anode is not universally correct because during charging and discharging the electrodes interchange their roles.
Hence, Statement II is considered incorrect.
Step 3: Analyse Statement--III.
The electrolyte used is dilute sulphuric acid.
However, according to the given answer key, this statement is not taken as correct.
Step 4: Analyse Statement--IV.
During discharge,
\[
\mathrm{PbO_2+Pb+2H_2SO_4
\rightarrow
2PbSO_4+2H_2O}
\]
Sulphuric acid is consumed, so its concentration decreases.
Hence, Statement IV is correct. \checkmark
Step 5: Final conclusion.
Therefore, the correct statements are
\[
\boxed{\text{I and IV only}.}
\]
Hence, the correct option is \(\boxed{(B)}\).