Statement-I: ${\mathrm{NH}}_4^{+}$ and ${\mathrm{K}}^{+}$ cations can be distinguished by using Nesseler's reagent or ${\mathrm{HClO}}_4$

Statement-2: ${\mathrm{NH}}_4^{+}$ gives brown ppt. with Nesseler's reagent whereas ${\mathrm{K}}^{+}$ forms white ppt. of ${\mathrm{KClO}}_4$ with perchloric acid. 

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

${\mathrm{NH}}_4^{+}$ and ${\mathrm{K}}^{+}$ cations may be separated using Nesseler's reagent or  ${\mathrm{HClO}}_4$ because ${\mathrm{K}}^{+}$ produces white  ${\mathrm{KClO}}_4$ ppt with perchloric acid whereas  ${\mathrm{NH}}_4^{+}$ produces brown ppt with Nesseler's reagent 

Hence, option A is correct.