"gerund " or "the present participle"

I’d like to know the part of speech of preparing which appears in the following sentence.

I’m busy preparing for tomorrow’s test.

I always appreciate your kind help.

‘Preparing…test’ is a participle clause complementing the adjective ‘busy’.

(1) The language coach has given you and me the answer.

(2) I only wish to point out most humbly that some people feel that there is a missing preposition. In other words: I am busy in preparing for tomorrow’s test. If you accept this theory, then “preparing” would be a gerund, the object of the understood preposition.

(a) Here are two examples from one scholar: (i) She has been busy a-ironing this evening. (ii) The miser has been busy a-hoarding his money.

(b) Many years ago, “a” was used as a preposition meaning “in,” “with,” “at,” etc. As the years passed, English speakers started dropping the “a.”

(3) One famous grammar book gives this example: I’m busy (with) getting the house redecorated. That book also agrees with the language coach: It says that the adjective “busy” is complemented by a “nominal -ing participle clause,” as in “They are busy preparing a barbecue.” That book (like some others) prefers not to use the term “gerund.”

I am busy in preparing for tomorrow’s test. (Unacceptable for me.)
I am busy preparing for tomorrow’s test. (Acceptable)
I am busy with preparing myself for tomorrow’s test. (My preference)

The comments please.

Thanks

Your feeling for the language is still embryonic.

Thank you everybody for your kind help.

My Japanese teacher also said that " in " is missing infront of preparing. So preparing is gerund.
But it was not enough for me to understand.
So I posted the question to Test Net.

You 've given me a lot of information.
Thank you. (^^)

If possible could you recomend some grammer books for me, please.

The only book that I can recommend is a big (1,779 pages) book entitled “A Comprehensive Grammar of the English Language” by Professor Randolph Quirk and his colleagues. Many language professionals use it; it is very difficult for ordinary people like me to understand. I have the 1985 edition. If you can get a copy, check out pages 1230 - 1231. It gives a good explanation about the use of “busy.”

I hope that the language professionals and other regular members can give you (and me) the names of newer (and easier) books that explain the use of “busy.”

As you already know, the Web can also help. I googled “Busy + gerund/ participle,” and I received many results in the “discussions” and “books” sections. Of course, you must be very careful about what you read on the Web. Since we ordinary people can give our opinions, many times ordinary people like me are completely wrong!!!

Thank you james M and everybody. I appreciate your kind help very much. (^^)

I thank you.