I feel like going for a walk.

Explanation: In this case, the phrase "feel like" is followed by a gerund form ("going") because it indicates a desire or inclination to do something. The gerund form is used after certain verbs, such as "like," "enjoy," "love," "hate," etc.

The exercise was done by Helen was completely correct.

Explanation: The sentence is not grammatically correct as it is. The correct form would be: "The exercise done by Helen was completely correct." Here, the verb "do" is in the past participle form ("done") because it acts as an adjective to describe the exercise. It indicates that Helen completed the exercise.

If they had done it yesterday, we would be glad.

Explanation: This sentence is in the third conditional form, which expresses a hypothetical situation in the past. In this case, the past perfect tense ("had done") is used in the if-clause to indicate an action that should have been completed before a certain point in the past. The main clause uses the conditional modal "would" and the base form of the verb ("be") to express the consequence of the hypothetical situation.

We just solved the problem.

Explanation: The verb "solve" is in the past simple tense ("solved") to indicate that the action of solving the problem occurred in the past. The adverb "just" indicates that the action happened a short time ago.

No mistakes were made by Joe at that lesson.

Explanation: The verb "make" is in the past participle form ("made") because it is used in the passive voice. The subject "Joe" is not performing the action; instead, the action is done to him. The passive voice is used here to emphasize that Joe didn't make any mistakes.