The further [A] / farther [B] he pushed himself, the more disillusioned he grew.

For the crowds it was more of a historical [A] / historic [B] event; for their leader, it was just another day.

The old man has a healthy distrust [A] / mistrust [B] for all new technology.

This film is based on a real [A] / true [B] story.

One suspects that the compliment [A] / complement [B] was backhanded

Use “farther” for physical distance and “further” for metaphorical, or figurative, distance. So further should be used here.
Correct usage for 2nd sentence is 'historic event', which means 'an event of great importance'.
In sentence 3, the word 'distrust', which means 'the feeling that someone or something cannot be relied upon' should be used.

The correct usage is sentence 4 is 'true' story.
A backhanded compliment, also known as a left-handed compliment or asteism, is an insult that is disguised as a compliment.

Option e) is the correct answer.