I have a book. The book is about physics.
To call a noun known or unknown (to put a/an OR the before it), I believe in this statement: an unknown object is something that the speaker knows but the audience does not. And a known object is something that both the speaker and the audience know.
First I wanted to know whether I'm right in finding it in my definition?
It's a test I found in 'Digest,' a grammar book I'm teaching in a class now.
Can you lend me .... pencil so that I can finish ..... test?
the answer is "a-the" but I think "a-a" can be correct too, am I right?
Yes, you are right in both cases.