I and II

The correct answer is option 2. I and II.

To determine which reactions are ways to synthesize carboxylic acids, let's analyze each option:

I. Oxidation of an Aldehyde

Aldehydes can be oxidized to carboxylic acids using oxidizing agents like potassium permanganate (\(\text{KMnO}_4\)), chromium trioxide (\(\text{CrO}_3\)), or nitric acid (\(\text{HNO}_3\)).

\(\text{RCHO} \longrightarrow \text{RCOOH}\)

This reaction is a common method for synthesizing carboxylic acids.

II. Hydrolysis of an Anhydride

Anhydrides react with water to form two molecules of carboxylic acids.

\((RCO)_2O + H_2O \longrightarrow 2 \text{RCOOH}\)

This reaction produces carboxylic acids and is a valid method for their synthesis.

III. Hydrolysis of a Ketone

Ketones do not hydrolyze to form carboxylic acids. Instead, ketones are generally stable compounds and do not undergo hydrolysis under normal conditions to yield carboxylic acids.

\(R_2CO + H_2O \longrightarrow \text{No reaction to form RCOOH}\)

Hydrolysis of a ketone does not lead to the formation of carboxylic acids.

Summary:

Oxidation of an aldehyde (I) is a way to synthesize carboxylic acids. Hydrolysis of an anhydride (II) is also a way to synthesize carboxylic acids. Hydrolysis of a ketone (III) does not synthesize carboxylic acids. Therefore, the correct answer is: I and II.