Step 1: Read the question.
We must name the enzyme that makes precursor mRNA in eukaryotes.
Step 2: Recall the three polymerases.
Eukaryotes use three RNA polymerases: I, II, and III, each transcribing different RNAs.
Step 3: Know what precursor mRNA is.
Protein-coding genes are first transcribed into heterogeneous nuclear RNA (hnRNA), which is the precursor (pre-mRNA) before processing into mature mRNA.
Step 4: Assign RNA polymerase I and III.
RNA polymerase I makes ribosomal RNAs (28S, 18S, 5.8S), and RNA polymerase III makes tRNA, 5S rRNA, and small RNAs. Neither makes pre-mRNA.
Step 5: Assign RNA polymerase II.
RNA polymerase II transcribes protein-coding genes into hnRNA, which is exactly the precursor mRNA we want.
Step 6: Conclude.
DNA polymerase copies DNA, not RNA, so it is ruled out. The answer is RNA polymerase II, which is option (3).
\[ \boxed{\text{RNA polymerase II}} \]