(i) Oxidative roasting of chromite with K$_2$CO$_3$ (air):
Chromite (\(\mathrm{FeCr_2O_4}\)) is oxidised to potassium chromate; iron becomes \(\mathrm{Fe_2O_3}\).
\[
\boxed{\,4\,\mathrm{FeCr_2O_4} + 8\,\mathrm{K_2CO_3} + 7\,\mathrm{O_2}
\;\longrightarrow\; 8\,\mathrm{K_2CrO_4} + 2\,\mathrm{Fe_2O_3} + 8\,\mathrm{CO_2}\,}
\]
(ii) \(\mathrm{K_2Cr_2O_7}\) + \(\mathrm{FeSO_4}\) in acidic medium:
Dichromate oxidises \(\mathrm{Fe^{2+}}\) to \(\mathrm{Fe^{3+}}\) and is reduced to \(\mathrm{Cr^{3+}}\).
\[
\boxed{\,\mathrm{Cr_2O_7^{2-}} + 6\,\mathrm{Fe^{2+}} + 14\,\mathrm{H^+}
\;\longrightarrow\; 2\,\mathrm{Cr^{3+}} + 6\,\mathrm{Fe^{3+}} + 7\,\mathrm{H_2O}\,}
\]
In terms of salts:
\[
\mathrm{K_2Cr_2O_7} + 6\,\mathrm{FeSO_4} + 7\,\mathrm{H_2SO_4}
\rightarrow \mathrm{K_2SO_4} + \mathrm{Cr_2(SO_4)_3} + 3\,\mathrm{Fe_2(SO_4)_3} + 7\,\mathrm{H_2O}
\]
(iii) Heating \(\mathrm{KMnO_4}\) to \(\sim 513\,\mathrm{K}\):
Permanganate disproportionates giving manganate, manganese dioxide and oxygen.
\[
\boxed{\,2\,\mathrm{KMnO_4} \;\xrightarrow{\,\Delta\,}\; \mathrm{K_2MnO_4} + \mathrm{MnO_2} + \mathrm{O_2}\,}
\]