Step 1: Calculate the moles of NaOH and HCl.
First, calculate the moles of NaOH and HCl:
\[
\text{Moles of NaOH} = 2.0 \, \text{M} \times 0.02 \, \text{L} = 0.04 \, \text{mol}
\]
\[
\text{Moles of HCl} = 0.2 \, \text{M} \times 0.05 \, \text{L} = 0.01 \, \text{mol}
\]
Step 2: Determine the excess NaOH.
Since NaOH is in excess, subtract the moles of HCl from NaOH, and then calculate the pH based on the concentration of excess OH\(^-\).
Step 3: Conclusion.
The final pH of the solution will be 12, as the excess OH\(^-\) makes the solution basic.
Final Answer:
\[
\boxed{12}
\]