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May I know which one is grammatical?

We will investigate on how different values of V affects the behaviour of the program.

We will investigate on how different values of V affect the behaviour of the program.

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  • william, it is not good or necessary, to mix American and English spellings in the same text. You should use one kind of spelling or the other, unless you include a name which uses just one kind.
    – Tristan
    Sep 2 '13 at 15:22
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Given that "different values of V" is plural (regardless of the plurality of _V itself), you would use affect.

We will investigate how different values of V affect the behaviour of the program.

Notice that I've removed on as that is not grammatically correct.

Another example could be the following one.

Seven breeds of dog are on show here today.

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