This is one of the most obvious additions. It's contradictory, it isn't even related to anything in its context, and ancient scribes couldn't even agree on where exactly it was supposed to go. Moreover, there's no old testament law about women being silent, so it's extremely unlikely that it was written by a Jew. One of the oldest copies in Codex Vaticanus has a mark on this passage warning that it's spurious and not in some copies. Codex Fuldensis has one too.

The other one is 1 Corinthians 11:3-16, which has many problems. First I'm going to point out that it could not have been located here in the original letter, based on its structure. The preceeding verses lead into verse 17, and this subject is a very sudden interruption. That doesn't necessarily mean Paul didn't write it and someone cut and pasted it here from another now-lost letter. But the language and theology suggests it was not him. On top of all this, it probably isn't actually sexist. It sounds that way, but then specifies "because of the angels". Even the church fathers admitted this: women were supposed to cover their heads to avoid tempting angels (with lust). Not as a status symbol. It actually concludes that women should not cover their heads as long as they have long hair. The last verse is usually mistranslated as "we have no other practice", but it actually says "we have no such practice". Whether this refers to material head coverings or rumors about Christian women not covering their heads is unclear. There's probably a lot of missing context there, as well as possible pagan rituals of this kind that the author didn't want Christians to engage in, especially seeing that angels and spiritual powers were involved. I don't think the author is quite happy with his explanation, because while affirming gender norms he makes sure to reassure gender equality in verse 11, unless that's another interpolation. Rather I think he's straining for an argument against material head coverings, and "long hair is already a covering" is the best he could do without flat-out rejecting the opposing argument. This could be seen as an attempt to compromise on an early debate about whether women need to cover their heads. And ancient people generally did not have an issue with such add-ons being made to their books, so they stuck the verdict right in one of the most popular ones so everyone would read it.

As a matter of fact a lot of 1 Corinthians is suspicious for the same reason, because the entire middle section seems to be trying to settle debates that would've arisen later, not merely in Corinth during Paul's time. "Paul" lays out some very detailed instructions almost like a rulebook, and seems to describe a church ahead of his time. And this is the longest letter besides Romans, both of which are waaaaaaaay too long for the first century, and Romans is full of massive interpolations. Also of note is how Marcion's version has almost no omissions in this letter, which might mean that it was redacted earlier than the others.