First things first, we notice commas in most of the answer choices. Commas are often a sign of modifier issues or problems with sentence structure, so we'll look out for that.

(A) Completing the physical examination, the tonsils were found to be diseased.

"Completing the physical examination" is a modifier - we need to find its subject. On the GMAT, a modifier's subject is always going to be the thing closest to it, which here is tonsils. So in this sentence, the tonsils were completing the examination. Tonsils can't examine anything, so we rule out A.

(B) Having completed the physical examination, the tonsils were found to be diseased.

Again, "Having completed the physical examination" is a modifier that needs a subject. And again, the closest subject is "tonsils". So the tonsils had completed the examination. Doesn't make sense. Eliminate B.

(C) When the physical examination was completed, the tonsils were found to be diseased.

This answer at least doesn't have a modifier error. We'll hang onto it for now.

(D) The physical examination completed was completed, the tonsils were found to be diseased.

Assuming that the first "completed" is a typo, we still have some pretty major issues with this answer choice. We should notice that both the part before the comma and the part after the comma are independent clauses - they can stand on their own as complete sentences.

• The physical examination was completed. The tonsils were found to be diseased.

This means that we are separating two independent clauses with only a comma, which is a comma splice and is grammatically incorrect. Eliminate.

(E) The physical examination found that the tonsils were diseased.

This is the most concise answer so far, and it makes perfect sense. Comparing it to C, the only answer we have left, we see not only that it is shorter (a good sign), but also that it makes the relationships in the sentence more clear. D tells us that the tonsils "were found to be diseased" after the physical examination, but it doesn't tell us what or who found them to be diseased - maybe the examination had nothing to do with it! E makes the causation clear - the examination found the disease.

So we eliminate C and pick E.