The underlined portion of the sentence contains a strange use of the verb "obsessing," which should not be used with the past tense "became." The correct answer choice must change "obsessing" to an appropriate past tense form; therefore, the correct answer is "The entire town became obsessed with the case."

The use of the verb "ignoring" is incorrect in this sentence, as it is a present participle and the sentence needs a past tense active verb; therefore, the correct answer is "Most people ignored."

The use of the past participle "keeping" is inappropriate as it needs either an infinitive or a version of "be" to be used as the verb in a sentence. "Keeping" needs to be changed to an appropriate verb form without changing the meaning of the sentence. The only answer choice that does both is "The man keeps up with."

The use of the verb form "questioning," a gerund, makes the sentence confusing. The word should be changed to a form that properly reflects the "scholars" are questioning "the power of monarchy." The answer choice that best does this is "is questioned by modern scholars."

The use of the participle form "keeping" is incorrect without a  version of the verb "to be," and needs to either be changed to a non-participle or have the version of "to be" added. This is the only issue with the sentence as it is written. Therefore, the correct answer choice is "keeps an extra pair of glasses."

The use of the participle form "moving" makes the sentence have no actual verb, as any participle needs a form of the verb "to be" to function as the verb in a sentence. The form needs to to either have the active form "is" attached or be changed into the active form; therefore, "The protesters move into position" is the correct answer choice.

The use of the verb "keeping," a participle, is incorrect and confusing in the sentence, as a participle always needs to function with a version of the form to be. Either one of these forms needs to be added in or the form should be changed. The only answer choice that appropriately does this is "the victorious nation kept the territory."

The underlined portion of the sentence needs to align with what it describes, the "scientists." To make it grammatically correct, the verb form of "debating" needs to be changed to a simple present form. "Still debate" is the correct answer choice.

The use of "making," a participle means the sentence has no verb, as a participle needs some form of "to be" to function as a full verb; therefore, a complete verb form needs to be used, and has to match the past tense usage of "tried." This makes the correct answer choice "made mistakes."

The use of "telling" in the sentence is strange and confusing, as the participle form appears to modify "ready." To appropriately explain what the father was "ready" for, an infinitive, which gives a sense of action; therefore, the correct answer choice is "to tell a joke."