In the following sentence, which of the following is used as a gerund or a gerund phrase? my sister was overjoyed about discovering a new hobby.

In the sentence above, "discovering" is used as a gerund.

Explanation:

Gerunds are words that are formed with verbs but act as nouns. Therefore, they can be subjects, subject complements, or objects in the sentence. In this case, the gerund "discovering" functions as the object of the preposition "about". Verbs after prepositions take the -ing form. As this word is derived from a transitive verb, it selects an NP object ( a new hobby).

recent brain research has shown some surprising benefits as a result of formal musical training. learning to play a musical instrument appears to develop parts of the brain that can even influence academic results. the studies indicate that the earlier a child begins training, the more benefits there are. one such study showed that children who began musical studies before the age of seven had a more developed brain in the region where language production and auditory processing take place. scientists believe there is mounting evidence to show that early musical training can improve brain function.

explanation:

given the fact that the question is about the detail that best support the idea that oppressive leaders use   propaganda to achieve thir goals,   the first option does not refer to any kind of propaganda, since it only tells about that the milk and the windfall apple should be reserved for the pigs. in the second option, that is, "snowball and napoleon sent out flights of pigeons whose instructions were to mingle with the animals on the neighbouring farms, tell them the story of the rebellion, and teach them the tune of "beasts of england". it can be identified that the oppressors make use of the propaganda, which is represented by the pigeons, and the information they have to tell other animals about what was happening in the farm in order to get their objectives. for that reason, the right answer is the second option, taking into account that the third option do not refer to propaganda since it only talks about that they did not want jones back. and, finally, the last option refers to the result of the given news, but it does not mean about the propaganda they used to achieve their goals.