13. (12 points) Four people, A, B, C, and D, participated in an exam with a full score of 100 points. Each person's score is an integer. After the exam, they predicted their scores and rankings:
A said: “My ranking is ahead of B, and also ahead of C.”
B said: “I scored 90 points, and I am 2 points higher than D.”
C said: “My ranking is ahead of D, and also ahead of A.”
D said: “I scored 91 points, and I am 3 points higher than B.”
After the results were released, it was found that each person's score was different, and each person's statements were half true and half false. Therefore, A scored ( ) points.
A. 90
B. 91
C. 92
D. 93
【Solution】Solution: According to the analysis, assuming the first half of Jia's statement is true and the second half is false, we get that Bing is in front of Jia, and through Bing, we know that the first half of his statement is false.

The obtained order is: Ding, Bing, Jia, Yi;
From this, we can determine that the first half of Yi's statement is true and the second half is false; the first half of Ding's statement is false, and the second half is true. From the table, we can see that Jia scores: 91 points.
\begin{tabular}{|c|c|c|c|}
\hline Ding & Bing & Jia & Yi \\
\hline 93 & 92 & 91 & 90 \\
\hline
\end{tabular}

Therefore, the answer is: $B$.