Key idea: boiling point depends mainly on intermolecular forces

For the given organic compounds, the important factors are:

• Hydrogen bonding
• Dipole–dipole interaction
• Molecular mass / van der Waals forces
• Branching

General trend among alcohols, ethers, and hydrocarbons

• Compounds with strong intermolecular hydrogen bonding have higher boiling points.
• Alcohols usually have higher boiling points than ethers of comparable molecular mass because alcohols can form intermolecular H-bonds.
• Among alcohols, more branching lowers boiling point because it reduces surface area and weakens van der Waals attraction.

Comparing the four compounds

From the standard trend intended in this question:

• (III) has the least intermolecular attraction, so it has the lowest boiling point.
• (IV) has slightly stronger intermolecular forces than (III), so its boiling point is higher.
• (II) shows stronger interaction than (IV), hence still higher boiling point.
• (I) has the strongest intermolecular hydrogen bonding / greatest effective attraction, so it has the highest boiling point.

Therefore, the increasing order is

Checking the options

• A: $(III) < (IV) < (II) < (I)$ ✅
• B: $(IV) < (III) < (I) < (II)$ ❌
• C: $(II) < (III) < (IV) < (I)$ ❌
• D: $(III) < (II) < (I) < (IV)$ ❌

Final answer

The correct increasing order of boiling points is

Hence, Option A is correct.