If roots of $a{x}^2+bx+c=0, a, b, c\in R, a\ne 0$ are imaginary, then 

Let $f\left(x\right)=a{x}^2+bx+c$.  If the roots of  $f \left(x\right)=0$ 

imaginary, then we cannot say anything about $b$ ( i.e. it can be

positive, negative or zero). So, options $\left(b\right), \left(c\right)$ and $\left(d\right)$ are not

necessarily true. 

Further, if $a> 0,$ then the graph of $y = f \left(x\right)$ is above x-axis and hence

$f\left(x\right)>0$ for all $x\in R\Rightarrow f\left(0\right)>0\Rightarrow c>0$

$\therefore ac>0.$

Similarly, if $a < 0,$ then the graph of $y = f \left(x\right)$ is below $x$-axis and hence

$f\left(x\right)<0$ for all $x\in R\Rightarrow f\left(0\right)<0\Rightarrow c<0$ 

$\therefore ac>0$