From proposition ①: a=b2+ 1.

→④

From proposition 2, we can get: 3 = a+b.

→⑤

From proposition ③: 0=4a+2b→⑥

If ① is a true proposition, then a=b2+ 1, that is, Equation ④ is an identity.

Substitute ④ into ⑤ and you will have a solution. Explain that ⑤ can be established, that is, proposition ② is true.

Substitute ④ into ⑥, and there is no solution. Explanation 6 cannot be established, that is, proposition 3 is false.

So choose the answer of "proposition 2 is true and proposition 3 is false", that is, option C.