In this excerpt form act i scene i of shakespeare’s richard iii what two purposes does richard soliloquy appear to serve?

The first purpose that Richard's soliloquy appears to serve is to reveal the motivations and personality of the character - we can see that he hates himself because he was basically a hunchback, so he was pretty self-conscious about it. He also reveals in the first scene what he intends to do throughout the play. 

As for the second purpose, I'd choose the answer - it informs the audience about the background of the plot, as it gives us reasons why Richard decided to do what he did. 

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how is the underlined gerund phrase used in the sentence?  

josh's plan, rushing the passer, seemed sound to us.

a.

object of a preposition

b.

direct object

c.

appositive

d.

subject

there are literally no sentences that i can say because there are no sentences attached to your question