Let  $\alpha ,\beta$  be the roots of  $x^2-x+p=0$  and  $\lambda ,\delta$  be the roots of  $x^2-4x+q=0$  such that  $\alpha ,\beta ,\gamma ,\delta$  are in G.P and  $p\ge 2.$  If  $a,b,c\in \{1,2,3,4,5\},$ let the number of equation of the form  $a{x}^2+bx+c=0$  which have real roots be r, then the value of 81 pqr =

$(\alpha +\beta )=1,\alpha \beta =p,\gamma +\delta =4,\gamma \delta =q$ 
Since $\alpha ,\beta ,\gamma ,\delta$ are in G.P
$\therefore \frac{\beta }{\alpha }=\frac{\delta }{\gamma }\Rightarrow \frac{\beta +\alpha }{\beta -\alpha }=\frac{\delta +\gamma }{\delta -\gamma }\Rightarrow \frac{{(\beta +\alpha )}^2}{{(\beta +\alpha )}^2-4\alpha \beta }=\frac{{(\delta +\gamma )}^2}{{(\delta +\gamma )}^2-4\delta \gamma }$ 
 $\Rightarrow \frac{1}{1-4p}=\frac{16}{16-4q}=\frac{4}{4-q}$
$\Rightarrow 4-q=4-16p$ 
Now,  $p\ge 2$   $\therefore q\ge 32$
For the given equation $a{x}^2+bx+c=0$  to have real roots   $b^2-4ac\ge 0$
$\therefore ac\le \frac{b^2}{4}$
 

Hence number of quadratic equations with real roots, r = 24
Now from (i) and (ii) the minimum value of   $pqr=\mathrm{2.32.24}=1536$