$f _1=\mu_1 N _1 $

$f _2=\mu_2 N _2$

In option $(A)(D)$ $\mu_1=0$

Net torque about $A$ should be zero

$ mg \cos \theta \frac{\ell}{2}= N _1 \sin \theta \ell $

$\Rightarrow N _1=\frac{ mg \cot \theta}{2} $

$\Rightarrow N _1 \tan \theta=\frac{ mg }{2} $

$\text { and } N _2= mg $

$\text { (B) } \mu_2=0$

There is no force to balance $N _1$ so rod can not remain in equilibrium

$\text { (C) } \quad $ $N _1=\mu_2 N _2 $

$N _2+\mu_1 N _1= mg $

$N _2+\mu_1 \mu_2 N _2= mg $

$N _2=\frac{ mg }{1+\mu_1 \mu_2}$