A) Use of 'are' is incorrect, does not agree with the singular subject 'the reason'.
B) Correct. Subject-verb matched correctly.
C) The verb 'has' should be used to refer 'the period' rather than 'have'.
D) Same as C.
E) Subject-Verb do not agree.

Has is singular and have is plural form.

He has a book.
They have pencils.

(A) has been established by carbon-14 dating, but what is much more difficult to determine
are (Correct SVA has been, but verb "are" does not match with "reason". Note, the subject and the verb are inverted here.
(B) has been established by carbon-14 dating, but what is much more difficult to determine
is (Correct Answer - has been matches with Period, and are matches with reason)
(C) (C)have been established by carbon-14 dating, but what is much more difficult to
determine is (Have been does not agree with Period)
(D) have been established by carbon-14 dating, but what is much more difficult to
determine are (Have been does not agree with Period)
(E) are established by carbon-14 dating, but which is much more difficult to determine is (A participle is required here, not a present tense. Additionally, are does not agree with Period. But which is incorrect...it appears to modify carbon-14 dating. Totally incorrect)

Regards
Jerry

grandh01 wrote: I am having difficulty with the two words has and have. I get confused in certain situations on usage. Can someone explain in detail the usage of the words has and have, thanks in advance.

I would be glad to help. Has is singular and have is plural. GMAT SC loves to trick us with subject-verb agreement. Remember, English sentences must agree in subject and verb. If you have a single subject you MUST have a single verb. Some examples:

Tom has a car. Tom is singular, has is singular.
They have a car. They is plural, have is singular.
Obviously, these two sentences are way too easy for the GMAT. So when you are doing these types of questions always locate the subject. In this problem the subject is THE PERIOD. Now is the period singular or plural? It's singular! You know tell because there is no s at the end of period and there's a the.

TAKE AWAY: ALWAYS look for the subject and then match the verb with it. Remember they try to trick you.

Let me know if you have more questions

grandh01 wrote:The period when the great painted caves at Lascaux and Altamlra were occupied by
Upper Paleollthle people has been established by carbon-14 dating, but what is much
more difficult to determine are the reason for their decoration, the use to which
primitive people put the caves, and the meaning of the magnificently depicted animals.
(A) has been established by carbon-14 dating, but what is much more difficult to determine
are
(B) has been established by carbon-14 dating, but what is much more difficult to determine
is
[/spoiler]

Can someone please explain me why A is wrong
Coming to the second part of the sentence, it is an inverted subject-verb pair

<the reason for their decoration>, <the use to which
primitive people put the caves>, and <the meaning of the magnificently depicted animals>

multiple nouns connected by and and we should plural verb - are -
what is much more difficult to determine are
can be reworded as
The things that Ram finds more difficult to determine are