The correct order of increasing $\mathrm{C}-\mathrm{O}$ bond length of CO, ${\mathrm{CO}}_3^{2-}, {\mathrm{CO}}_2 \mathrm{is}$

Bond length $\propto \frac{1}{\mathrm{Bond} \mathrm{order}}$

Bond order : ${\mathrm{CO}}_3^{2-} < {\mathrm{CO}}_2 < \mathrm{CO}$

Bond order in CO = 3 (with the help of molecular orbital theory)

Bond order in ${\mathrm{CO}}_2=\frac{\mathrm{No}. \mathrm{of} \mathrm{bonds} \mathrm{in} \mathrm{all} \mathrm{possible} \mathrm{sides}}{\mathrm{No}. \mathrm{of} \mathrm{resonating} \mathrm{structure}}$

(by resonance)               $=\frac{4}{2}=2$

Bond order in

 $$\begin{gathered} {\mathrm{CO}}_3^{2-}=\frac{4}{3}=1.33 (\mathrm{by} \mathrm{resonance}) \\ \mathrm{So} \mathrm{order} \mathrm{to} \mathrm{bond} \mathrm{length} \mathrm{of} \mathrm{C}-\mathrm{O} \\ \mathrm{CO} < {\mathrm{CO}}_2 < {\mathrm{CO}}_3^{2-}. \end{gathered}$$