Step 1: Recall the rule for paramagnetism.
A species is paramagnetic if it has one or more unpaired electrons; otherwise, it is diamagnetic.
Step 2: Write d–electron configurations.
Ag$^{+}$ : $4d^{10}$ (all paired) $\Rightarrow$ diamagnetic.
Cu$^{2+}$ : $3d^{9}$ (one unpaired) $\Rightarrow$ paramagnetic.
Zn$^{2+}$ : $3d^{10}$ (all paired) $\Rightarrow$ diamagnetic.
Au$^{+}$ : $5d^{10}$ (all paired) $\Rightarrow$ diamagnetic.
Step 3: Conclusion.
Only Cu$^{2+}$ has an unpaired electron; hence it is paramagnetic.