Two non-conducting solid spheres of radii R and 2R , having uniform...

Field due to small sphere
$= \dfrac{\text{K} {ρ}_{1} \dfrac{4}{3} \pi {\text{R}}^{3}}{{\left(2 \text{R}\right)}^{2}}$
Field due to bigger sphere
$= \dfrac{\text{K} {ρ}_{2} \dfrac{4}{3} \pi {\left(2 \text{R}\right)}^{3} \times \text{R}}{{\left(2 \text{R}\right)}^{3}}$
The net electric field at point $P$ will be

$\overset{\to}{E}net = 0$

Equating above two, we get $\dfrac{{ρ}_{1}}{{ρ}_{2}} = 4$
[Charges cannot be of an opposite sign otherwise field will not zero]

Similarly, the net electric field at point $Q$ becomes zero. Then $\dfrac{{ρ}_{1}}{{ρ}_{2}} = - \dfrac{3 2}{2 5}$ here both spheres should have opposite polarity.