Step 1: The strength of the acid is determined by the ability of the molecule to donate a proton (H\(^+\)). The electron-withdrawing groups, such as \( \text{NO}_2 \), enhance the acidic strength by stabilizing the conjugate base. The electron-donating groups, such as \( \text{OH} \), make the molecule less acidic.
Step 2: Arranging based on acidic strength:
- Option (IV) \( \text{C}_6 \text{H}_4 (\text{NO}_2)(\text{OH}) \): The \( \text{NO}_2 \) group, being highly electron-withdrawing, increases the acidity.
- Option (III) \( \text{C}_6 \text{H}_4 (\text{NO}_2)(\text{OH}) \): The \( \text{NO}_2 \) group is still electron-withdrawing, but slightly less than in option IV.
- Option (I) \( \text{C}_6 \text{H}_5 (\text{OH}) \): The \( \text{OH} \) group is electron-donating, which decreases the acidity.
- Option (II) \( \text{C}_6 \text{H}_4 (\text{OH}) \): This is the least acidic because of the electron-donating effect of \( \text{OH} \).
Thus, the order of acidic strength is IV > III > I > II.