b) c) and d) are wrong. If I'm honest, I've not heard "go down + with" used like this much in my life, and I guess the sources would have been American. It is colloquial and not even one of the most common uses of "go down". "down" feels right because of the comparison between digesting food and believing something difficult. Food goes "down" inside you when you digest it. So here, the idea is that the truth cannot be digested. I'd say it would be quite an unusual meaning of a phrasal verb to focus on in a test question, unless it's actually really common in the USA or elsewhere. A quick check of a sentence database didn't reveal anything useful.

This is super-colloquial, I'm surprised it's on a quiz. Michael is absolutely correct, in the US "going down" can mean "taken in" in the sense of ingesting (swallowing) something, but more often it means "happening": - It's a shame the way it all went down. <= the way it happened

I agree with Michael, John and James. As with the other members, my native-speaker instinct told me that whoever wrote this quiz intended the answer to be 'down', in the sense of 'won't take me in/''won't fool me', but it doesn't feel at all natural. I've also checked various sentence databases and I can't find a single use of 'go down with someone' with this meaning. The nearest common expression is 'go down well with someone' which means 'receive a favourable reception', as in 'It won't go down well with your boss when you tell him you've lost the contract'. If this quiz came from a Russian site, I suspect it was written by a non-native English speaker who was determined to find an example of 'go down with' and didn't quite get it right. If it's an online source, feel free to post the link and we can give our opinion on whether it's reliable or not.

It's probably "down", which you perhaps more often see with an adverb and without an object, as in "Did your presentation go down well [with the audience], dear?" However, I don't altogether trust the quiz question here, given the comma misuse in it.