Step 1: Identify isomerism in \([Pt(NH_3)_2Cl_2]\).
This complex has two ammonia ligands and two chloride ligands.
Such square planar complexes can show cis-trans forms.
So:
\[
[Pt(NH_3)_2Cl_2]
\]
shows:
\[
\text{Geometrical isomerism}.
\]
Therefore:
\[
A-III
\]
Step 2: Identify isomerism in \([Co(en)_3]Cl_3\).
The ligand \(en\) is ethylenediamine.
It is a bidentate ligand.
Complexes having three bidentate ligands can exist as non-superimposable mirror images.
Therefore:
\[
[Co(en)_3]Cl_3
\]
shows:
\[
\text{Optical isomerism}.
\]
So:
\[
B-I
\]
Step 3: Identify isomerism in \([Co(NH_3)_5NO_2]Cl_2\).
The ligand \(\ce{NO2^-}\) can coordinate through nitrogen or oxygen.
If it coordinates through nitrogen, it is called nitro.
If it coordinates through oxygen, it is called nitrito.
This gives linkage isomerism.
So:
\[
C-IV
\]
Step 4: Identify isomerism in \([Cr(H_2O)_6]Cl_3\).
Water molecules may be present inside or outside the coordination sphere in hydrated complexes.
This gives solvate or hydrate isomerism.
So:
\[
D-II
\]
Step 5: Final matching.
Thus, the correct matching is:
\[
A-III,\quad B-I,\quad C-IV,\quad D-II.
\]