The correct answer is A.
a. National Highway that passes through Bihar: National Highway No. 12 passes through Bihar.
b. National Highway that does not pass through Bihar: National Highway No. 87 does not pass through Bihar.
c. National Highway having maximum length in Bihar: National Highway No. 2 has the maximum length in Bihar.
d. Proposed National Highway not to serve Bihar: National Highway No. 31 is a proposed national highway that is not to serve Bihar.
Here is a table showing the details of the national highways mentioned in the question:
| National Highway | Number | Starts from | Ends at | Length in Bihar |
| — | — | — | — | — |
| National Highway No. 12 | 75 | Kanpur, Uttar Pradesh | Barauni, Bihar | 240 km |
| National Highway No. 87 | 87 | Varanasi, Uttar Pradesh | Forbesganj, Bihar | 200 km |
| National Highway No. 2 | 2 | Agra, Uttar Pradesh | Kanyakumari, Tamil Nadu | 2,465 km |
| National Highway No. 31 | Proposed | Varanasi, Uttar Pradesh | Kolkata, West Bengal | 1,000 km |