Step 1: Expected behaviour of aldehydes.
Open-chain glucose contains an aldehyde group ($-CHO$), which should react with 2,4-dinitrophenylhydrazine to form a hydrazone.
Step 2: Actual observation.
Glucose does not give the 2,4-DNP test, indicating the absence of a free aldehyde group in its predominant form.
Step 3: Reason for this behaviour.
The aldehyde group of glucose reacts intramolecularly with a hydroxyl group to form a hemiacetal ring structure, resulting in a cyclic form of glucose.
Step 4: Conclusion.
The failure of glucose to react with 2,4-dinitrophenylhydrazine confirms that glucose mainly exists in a cyclic form.