Concept:
This question is based on aldol condensation reactions of carbonyl compounds and characteristic reactions of ketones.
Step 1: Understanding the reaction of acetone with base.
Acetone contains \(\alpha\)-hydrogen atoms and therefore undergoes aldol condensation in the presence of a dilute base such as \( \mathrm{Ba(OH)_2} \).
Initially, two molecules of acetone combine to form diacetone alcohol:
\[
2CH_3COCH_3 \xrightarrow{\mathrm{Ba(OH)_2}} CH_3COCH_2C(OH)(CH_3)_2
\]
The product formed is 4-hydroxy-4-methylpentan-2-one.
Step 2: Dehydration on heating.
On heating, the aldol product loses one molecule of water and forms an \(\alpha,\beta\)-unsaturated ketone called mesityl oxide.
\[
CH_3COCH_2C(OH)(CH_3)_2 \xrightarrow{\Delta} CH_3COCH=C(CH_3)_2
\]
The IUPAC name of this compound is:
\[
\text{4-Methylpent-3-en-2-one}
\]
Thus, statement (1) is correct.
Step 3: Checking the other options.
Acetone with \( \mathrm{NH_2NH_2/KOH} \) undergoes Wolff–Kishner reduction and forms propane, not butane.
Acetophenone can be prepared from benzoyl chloride using dimethyl cadmium.
Acetophenone contains the \( \mathrm{CH_3CO-} \) group and therefore gives a positive iodoform test.
Hence, all remaining statements are incorrect.