[black_lily]

to me, 1a sounds more natural...as to second group of sentences, none of them sound to me correct...I like susan53's examples...in the third group, the correct answer is a...regarding the fourth group, I'm not quite sure but I'd go with b

[susan53]

Sorry Lily but I don't agree. in the third group all of them are "correct" but (3a) and (3c) are much more likely to be heard than (3b)

Generally, when using There + Be, the verb agrees with the first item to be mentioned, regardless of what comes later. So, even though 3b is grammatically logical, it would be unlikely to be used. We'd more likely say one of the following :

There's a book on the table.
There's a book and five pencils on the table.
There are five pencils and a book on the table.

(NB notice the contraction which makes the first two sentences sound more natural)

In exactly the same way, this makes 4a much more likely than 4b, which sounds very unnatural.

In really natural speech though, it would be unlikely to be expressed like this - information tends to be "chunked" to make it easier for the speaker to think of what they want to say and the listener to process it. So it would be more likely to be:

A : I need something to write with.
B : There's a pen on the table - oh and some pencils too if you prefer.

A : I need something to write with.
B : There's a pencil on the table - oh and a pen too if you prefer.