Rate of electrophilic substitution in substituted anilines and Steric Inhibition of Resonance (SIR effect).
LOGIC:
1. Coupling with $PhN_2Cl$ is an electrophilic aromatic substitution.
2. Reactivity increases with electron density on the ring. The $-NMe_2$ group is strongly activating (+M).
3. In compound R, the two ortho-methyl groups cause 'Steric Inhibition of Resonance'. The $-NMe_2$ group rotates out of the ring plane, stopping the lone pair from entering the ring. Hence, R is least reactive.
4. In P, the resonance is maximum as there is no hindrance.
5. In Q, the meta-methyl provides a small +I effect but does not interfere with the resonance of the amino group as much as ortho groups do, though P remains more accessible.
6. Thus, the rate order is P > Q > R. Option (1).