Let's check between C and E.The application of a new method for analyzing the chemistry of tooth enamel allows scientists to examine molars of prehuman ancestors and determining
that their diets were more varied than had been supposed.
Consider the construction, when they applied a method, they deduced that molars of ...
When you apply a method, you deduce something - the result of an experiment.
Compare the same with option 'C'
(C) When they applied a new method
for analyzing the chemistry of tooth enamel, scientists examined molars
of prehuman ancestors, determining
'C' is awkward.By applying a method, they examined molars ...
is a better construction.
The next error is with the usage of 'determining' which is acting as noun-ing modifier, modifying (correctly) scientists. No issues with this. But as a consequence, 'that their diets...' also modifies 'scientists' - subject of the preceding clause rather than 'ancestors' - object of the preceding clause.
Note that a pronoun need not always refer to the closest noun. Additionally, 'When they applied ... , scientists... , that their ...' also proves that the pronouns are referring to the same subject - scientists, ambiguously.
All these issues are corrected in 'E'. Underlined the modifiers and subjects for you (always remove the prepositional phrases to see how the subjects and modifiers are used)
(E) Applying a new method
for analyzing the chemistry of tooth enamel, scientists have examined
molars of prehuman ancestors and determined
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What are modifiers ??