Concept:
This question matches important Indian sculptural themes and figures with their archaeological or artistic sites.
Step 1:
The Mahakapi Jataka narrative is associated with:
\[
Bharhut
\]
Thus:
\[
A \rightarrow II
\]
Step 2:
The famous bronze Dancing Girl sculpture belongs to:
\[
Mohenjo Daro
\]
Thus:
\[
B \rightarrow IV
\]
Step 3:
The Chakravartin representation is associated with:
\[
Jaggayyapeta
\]
Thus:
\[
C \rightarrow I
\]
Step 4:
The monumental Varaha sculpture is located at:
\[
Udayagiri
\]
Thus:
\[
D \rightarrow III
\]
Therefore, the correct matching is:
\[
A-II,\; B-IV,\; C-I,\; D-III
\]
Hence, the correct answer is:
\[
\boxed{\text{(1) A-II, B-IV, C-I, D-III}}
\]