A block of mass ‘m’ is kept on a smooth movable wedge. If the acceleration of the wedge is ‘a’

The acceleration of block w.r.t wedge is given by 

$a_r=g\sin \theta -a\cos \theta$

The net acceleration is the resultant of the acceleration of wedge and acceleration of block w.r.t wedge

$\begin{array}{l}{a}_{\text{net }}=\sqrt{a^2+a_r^2+2a\left(a_r\right)\cos \theta }\\ =\sqrt{a^2+(g\sin \theta -a\cos \theta {)}^2+2a(g\sin \theta -a\cos \theta )\cos \theta }\\ =\sqrt{a^2+g^2{\sin }^2\theta +a^2{\cos }^2\theta -2ga\sin \theta \cos \theta +2ga\sin \theta \cos \theta -2a^2{\cos }^2\theta }\end{array}$

$\begin{array}{l}=\sqrt{a^2+g^2{\sin }^2\theta -a^2{\cos }^2\theta }\\ =\left(\sqrt{a^2+g^2}\right)\sin \theta \end{array}$

Also $N=mg\cos \theta +ma\sin \theta$

$\begin{array}{l}\therefore N_1=Mg+N\cos \theta \\ =Mg(mg\cos \theta +ma\sin \theta )\cos \theta \\ =Mg+mg{\cos }^2\theta +ma\sin \theta \cos \theta \end{array}$

$F_{ext}-N\sin \theta =Ma$

$\begin{array}{l}{F}_{ext}=Ma+(mg\cos \theta +ma\sin \theta )\sin \theta \\ =Ma+mg\sin \theta \cos \theta +ma{\sin }^2\theta \end{array}$