In the following sentence I have to find the error
The ship sank and everyone abroad drowned.
I think instead of abroad onboard should be used, and it is the only error. But the answer given in my book is
In the place of drowned, was drowned should be used. But I think its incorrect.
I think you mean to ask about aboard and onboard. “Abroad” means something rather different.
Alternatively, perhaps you have misread aboard as abroad. Have a look at your dictionary. They are different words.
Personally, I find onboard abhorrent, and on board slightly less so. Aboard is a great word and deserves more attention, like its kin abaft, abeam, athwart, and astern, as well as above, about, around, along, and, yes, abroad, and probably others I’m not thinking of just now.
As to the assertion that the sentence should read was drowned instead of drowned, that is not correct. It might be correct if your goal is to sound like someone who was writing in the early 19th century or earlier. Either drowned or was drowned would be correct in that sentence, but drowned is rather more normal for modern English.
Both are possible because drown has both transitive and intransitive sentences. Intransitively, it means to be suffocated in liquid (usually water):
The person drowned in the lake. Witnesses heard cries of “Help, I’m drowning!”
Transitively, it means to suffocate something or someone in liquid (usually water):
The murderer drowned the victim in the lake. Witnesses heard cries of “Help, I’m being drowned!”