Concept:
According to Brønsted–Lowry theory:
• A conjugate acid is formed by addition of a proton (\(H^+\)).
• A conjugate base is formed by removal of a proton (\(H^+\)).
Step 1: Finding the conjugate acid of \(\mathrm{HCO_3^-}\).
Add one proton:
\[
\mathrm{HCO_3^- + H^+ \rightarrow H_2CO_3}
\]
Thus, the conjugate acid is:
\[
\mathrm{H_2CO_3}
\]
Step 2: Finding the conjugate base of \(\mathrm{HCO_3^-}\).
Remove one proton:
\[
\mathrm{HCO_3^- \rightarrow CO_3^{2-}+H^+}
\]
Thus, the conjugate base is:
\[
\mathrm{CO_3^{2-}}
\]
Therefore, the correct answer is:
\[
\boxed{(B)\ \mathrm{H_2CO_3\ and\ CO_3^{2-}}}
\]