The term meta-directing refers to substituents on a benzene ring that direct incoming electrophiles to the meta position (relative to themselves) during an electrophilic aromatic substitution reaction.

Meta directors are typically deactivating groups with respect to the aromatic ring and they often posses an atom with a partial positive charge or a full formal positive charge directly attached to the ring, which tends to withdraw electron density from the ring through the inductive effect or resonance. This decreased electron density in the ring makes the ortho and para positions less reactive, thereby directing new substituents to the meta position. Furthermore, the presence of these withdrawing groups can stabilize the intermediate cation (whichever is formed) better when the substituent is at the meta position relative to the ortho or para positions.

Let's examine the options:

Option A (Incorrect):

• $-\mathrm{CN}$ (cyanide) is a strong meta director due to its strong electron-withdrawing nature.
• $-\mathrm{NH}_2$ (amino), $-\mathrm{NHR}$ (alkylamino), and $-\mathrm{OCH}_3$ (methoxy) are strong ortho/para directors because they donate electron density to the ring through resonance.

Option B (Possibly correct):

• $-\mathrm{CN}$ (cyanide) is a strong meta director as stated before.
• $-\mathrm{CHO}$ (formyl) is a weak meta director because the carbonyl group withdraws electron density via resonance and induction.
• $-\mathrm{NHCOCH}_3$ (acetamido) is a bit tricky. While $-\mathrm{NH}_2$ usually directs to the ortho/para position, when it is part of the amide group, the resonance with the carbonyl can make it withdraw electron density, thus Acetamido become a meta-director.
• $-\mathrm{COOR}$ (ester) is a meta director due to the electron-withdrawing nature of the carbonyl group.

Option C (Incorrect):

• $-\mathrm{NO}_2$ (nitro) is a very strong meta director due to its strong electron-withdrawing nature both through induction and resonance.
• $-\mathrm{NH}_2$ (amino) and $-\mathrm{COOH}$ (carboxylic acid) are strong ortho/para directors. Carboxylic acid groups can be deactivating due to their overall electron-withdrawing nature, but they still direct to the ortho/para positions because the electronegative oxygen can donate electron density by resonance.
• $-\mathrm{COOR}$ (ester) is a meta director.

Option D (Correct):

• $-\mathrm{NO}_2$ (nitro) is a strong meta director as before.
• $-\mathrm{CHO}$ (formyl) is a meta director.
• $-\mathrm{SO}_3 \mathrm{H}$ (sulphonic acid) is a strong meta director because the sulphonic acid group is very electron-withdrawing due to the presence of several electronegative oxygen atoms.
• $-\mathrm{COR}$ (ketone) is a meta director because of the electron-withdrawing nature of the carbonyl group.

Based on the description above, the set of functional groups that are meta directing from the given options is Option D: $-\mathrm{NO}_2$, $-\mathrm{CHO}$, $-\mathrm{SO}_3 \mathrm{H}$, $-\mathrm{COR}$. These groups are all electron-withdrawing and would direct incoming substituents to the meta position during electrophilic aromatic substitution.