Titration of sodium carbonate and nitric acid
To solve this problem, you need to know the pKas of carbonic acid.

The pKas are 6.35 and 10.33.

If you add 1 equivalent of acid to $\ce{Na_2CO_{3}}$, the product will be $\ce{NaHCO_{3}}$ (sodium bicarbonate).

In this case, you are going 0.4-.377 = 0.023 moles past the point of 1 equivalent.  

I would model the final solution as 0.354 moles of $\ce{HCO_{3}^-}$ and 0.023 moles $\ce{H_2CO_{3}}$ , and use the pKa of 6.35 and the Henderson-Hasselbalch equation to calculate pH.

The proposed answer 10.77 is incorrect.