(a) ${\mathrm{CoCl}}_3.4{\mathrm{NH}}_3$,

(b) ${\mathrm{CoCl}}_3·5{\mathrm{NH}}_3$,

(c) ${\mathrm{CoCl}}_3·6{\mathrm{NH}}_3$ and

(d) $\mathrm{CoCl}{\left({\mathrm{NO}}_3\right)}_2·5{\mathrm{NH}}_3$

Number of complex(es) which will exist in cis-trans form is/are

Cis-trans isomerism is a type of geometrical isomerism.This type of isomerism occurs when there is a change in spatial arrangements of ligands. Cis and trans are the two terms used for representing the position of identical ligands when they are placed adjacent and opposite

(i) ${\mathrm{CoCl}}_3.4{\mathrm{NH}}_3\Rightarrow \left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]\mathrm{Cl}$, can exhibit cis-trans isomerism.

(ii) ${\mathrm{CoCl}}_3.5{\mathrm{NH}}_3\Rightarrow \left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_5\mathrm{Cl}\right]{\mathrm{Cl}}_2$,can't exhibit cis-trans  isomerism.

(iii) ${\mathrm{CoCl}}_3.6{\mathrm{NH}}_3\Rightarrow \left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_6\right]{\mathrm{Cl}}_3$,can't exhibit cis-trans  isomerism.

(IV)

 $$\begin{gathered} \mathrm{CoCl}{\left({\mathrm{NO}}_3\right)}_2.5{\mathrm{NH}}_3\Rightarrow \left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_5\mathrm{Cl}\right]{\left({\mathrm{NO}}_3\right)}_2 \\ \mathrm{or} {\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_5{\mathrm{NO}}_3\right]}_{}\mathrm{Cl} {\mathrm{NO}}_3 \end{gathered}$$

Both can't exhibit cis-trans  isomerism.

Hence. only one complex exhibits cis-trans isomerism