Consider the given diagram –
Moment of force due to F at A (anticlockwise)
\(τ_1=\vec{AB}\times \vec{F}=a\times \vec{F} \)
Moment of force due to mg at A (Clockwise),
τ2 = mg × \(\frac{a}{2}\)
(i) Cube will not move (exhibit motion), if τ1 = τ2
\(a\times \vec{F}\) = mg\(\frac{a}{2}\) or \(\vec{F}\) = \(\frac{mg}{2}\) so, (a) → (ii)
(ii) Cube will rotate, if τ1 > τ2
Fa > mg\(\frac{a}{2}\) or F > \(\frac{mg}{2}\) So, (b) → (iii)
(iii) Let Normal reaction is acting at a/3 from point A,
mg× \(\frac{a}{3}\) = F × a or F = \(\frac{mg}{3}\) > \(\frac{mg}{4}\) So, (c) → (i)
(iv) \(\frac{mg}{4}<\frac{mg}{3}\) if, so due to F, block will not move, so, (d) → (iv)
∴ (a) → (i), (b) → (iii), (c) → (i), (d) → (iv)