Agreement between subject and verb in a number of / the number of [duplicate]

3 sentences:-

1. A number of such incidents has/have been reported by the local residents so far.
2. The number of such incidents is/are very low.
3. The number of elephants in Kerala has/have been dwindling.

What is the correct form of verb in the sentences, and why?

[1] A number of such incidents has/have been reported by the local residents so far.

[2] The number of such incidents is/are very low.

[3] The number of elephants in Kerala has/have been dwindling.

In [1] "number", with the indefinite article "a", is a non-count quantification noun that is 'number transparent' in that the number of the whole NP is determined by the number of the NP (the 'oblique') that is complement of the preposition "of". As it happens, "number" permits only plural obliques, like "incidents", and hence the correct verb is the plural "have".

The number-transparent use of "number" in [1] indicates an imprecise number, but in [2] and [3] the definite article indicates a precise number, where the subject NP is singular by virtue of having singular non-transparent "number" as head. Thus, the correct verbs in [2] and [3] are singular "is" and "has".