*The brown shirt 'wants washing'.
- has to wash
- is in need of a wash
- requires a wash
- no improvement.
This is my exam question. In provisional answer key the answer has given 3. But what about options no. 2 & 4?
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The fact of the matter is that several of the answers provided are correct, so it's impossible to explain why the test answers key indicates that only one is correct (and it's not the one that is currently the most widespread one, according to the ngrams).
The reason for not being able to make hands or tails of the author's reasoning is that the author of the testbook apparently uses his own version of English (a mild way of putting it)—details below. This version of English is gravely out of alignment with the mainstream English-speaking countries. So I doubt anyone here will be able to help the OP with the reasoning behind this book—this is not a logic that would be valid for mainstream English. The testbook is bound to be riddled with errors—errors in how the tests are put together, since basically almost of all the provided sentences are correct, but the test logic requires the user to consider almost all of them as incorrect; so this is an error in the test, and if there is one such error (and given that what I've seen from the book introduction is written in extremely clumsy English), there will be more errors and "correct answers" that cannot be explained.
This test is taken from a book called "Topic-wise Latest 43 Solved Papers", which already sounds fishy. The description contains phrases like "papers from 2010 to 2017 which have been provided topic-wise along with detailed solutions." This is not correct English as used by native speakers. If you google "provided topic-wise", you get 443 results, mostly of this same Indian book. If you google for the correct English phrase "arranged topically", you get 177 thousand results—and also 346 thousand results for "arranged by topic".
The book description further states, "Aspirants will come to know about the pattern and toughness of the questions asked in the examination. In the end, this book will make the aspirants competent enough to crack the uncertainty of success in the Entrance Examination."
OP, I guess you'll just have to get cracking at the uncertainty, because the author of your testbook has their own version of English.
It is a sad fact that incompetent teachers of English write books, too.
I sincerely sympathize.
You'd have to really just study up on the thought process of this particular author. If it is possible, try to ask your teacher or school/college to pick a different exam book, or team up with TOEFL, or something.
The specific question about the shirt has been asked before on wordreference, and, per native speakers who have answered it there, the "title phrasing" (thus, answer 4) and answers 2 and 3 are correct. It is also proven by ngrams of these phrases.