Answer (b)
Presence of electron withdrawing group on phenols, increases its acidic strength. So, both compounds i.e., $p$-nitrophenol (II) and $m$-nitrophenol (IV) are stronger acid than (I). If this $-NO_2$ group is present at $p$-position, then it exerts both $-I$ and $-R$ effect but if it is present at meta position, then it exerts only $-I$ effect. Therefore, $p$-nitrophenol is much stronger acid then $m$-nitrophenol.
On the other hand, presence of electron releasing group on phenol, decreases its acidic strength. If $-OCH_3$ group is present at meta position, it will not exert $+R$ effect but exert $-I$ effect.
But, if it is present at para position, then it will exert $+R$ effect. Therefore, $m$-methoxy phenol is more acidic than $p$-methoxy phenol.
Hence, the correct option is (b).