With what acceleration 'a' should the box of figure moving up so that the block of mass M exerts a force 7 Mg/4 on the floor of the box?

FBD of M:  If M exerts force F : 7$\frac{\mathrm{Mg}}{4}$ on floor, then from third law floor also exerts force F on box in upward direction.

F - Mg = Ma
$\Rightarrow \frac{7\mathrm{Mg}}{4}-\mathrm{Mg} = \mathrm{Ma} \Rightarrow \mathrm{a} = \frac{3\mathrm{g}}{4}$