1. Interpret the composition

   The complex has molecular composition: $\mathrm{Co}\cdot 5\mathrm{NH}_3\cdot \mathrm{Cl}_2\cdot \mathrm{SO}_4$

   Since it is an octahedral cobalt complex with 6 coordination positions, and there are 5 neutral ligands $\mathrm{NH}_3$, the 6th position must be occupied by either:

   • one $\mathrm{Cl^-}$, or
   • one $\mathrm{SO_4^{2-}}$ group (as coordinated sulfate).

2. Use the precipitation tests

   • Isomer A gives white precipitate with $\mathrm{AgNO_3}$.

     This indicates presence of free chloride ion in solution, because: $\mathrm{Ag^+ + Cl^- \rightarrow AgCl(s)}$

   • Isomer B gives white precipitate with $\mathrm{BaCl_2}$.

     This indicates presence of free sulfate ion in solution, because: $\mathrm{Ba^{2+} + SO_4^{2-} \rightarrow BaSO_4(s)}$

3. Write the two isomeric forms

   Therefore, the two isomers must be:

   • A: free $\mathrm{Cl^-}$ outside the coordination sphere, sulfate coordinated inside: $[\mathrm{Co}(\mathrm{NH}_3)_5\mathrm{SO}_4]\mathrm{Cl}$

   • B: free $\mathrm{SO_4^{2-}}$ outside the coordination sphere, chloride coordinated inside: $[\mathrm{Co}(\mathrm{NH}_3)_5\mathrm{Cl}]\mathrm{SO}_4$

4. Identify the isomerism

   Here, the ligand inside the coordination sphere and the counter ion outside the coordination sphere are interchanged: $\mathrm{Cl^-} \leftrightarrow \mathrm{SO_4^{2-}}$

   This is the definition of ionisation isomerism.

5. Check options

   • A: Co-ordinate isomerism — Incorrect
   • B: Ionisation isomerism — Correct
   • C: Geometrical isomerism — Incorrect
   • D: Linkage isomerism — Incorrect

Final Answer

$\boxed{\text{B: Ionisation isomerism}}$