To determine chirality, we check whether a molecule has a plane of symmetry or not. A molecule is chiral if it lacks any plane of symmetry and has at least one chiral center. A molecule with a plane of symmetry is achiral.
Molecule I: This molecule has a symmetrical substitution on the cyclohexane ring, with the methyl and hydroxyl groups in equivalent positions. There exists a plane of symmetry passing through the ring, making it achiral.
Molecule II: Similar to molecule I, the methyl and hydroxyl groups are placed symmetrically on the cyclohexane ring. Hence, a plane of symmetry exists, making this molecule also achiral.
Molecule III: Although there are two chiral centers, the molecule is symmetric. It contains an internal mirror plane, which divides the molecule into two equal halves. Such a molecule is a meso compound, which is achiral despite having chiral centers.
Molecule IV: This molecule has two chiral centers but lacks a plane of symmetry. The spatial arrangement of groups is asymmetric, which makes this molecule chiral.
Thus, only I and II are achiral, and IV is chiral.