If the equation ${\mathrm{ax}}^2+2\mathrm{bx}+3\mathrm{c}=0$ and $3{\mathrm{x}}^2+8\mathrm{x}+15=0$ have a common root, where a,band c are the lengths of the sides of a$\mathrm{△}\mathrm{ABC},$ then ${\sin }^2\mathrm{A}+{\sin }^2\mathrm{B}+{\sin }^2\mathrm{C}$ is equal to

Discriminant o{ the equation $3{\mathrm{x}}^2+8\mathrm{x}+15=0$ is given by
$\mathrm{D}=64-180=-116<0$
so  it , roots are imaginary and therefore roots are conjugate to each other. Therefore, one common root means both the roots are common.
$\therefore \frac{\mathrm{a}}{3}=\frac{2\mathrm{b}}{8}=\frac{3\mathrm{c}}{15}$
$\Rightarrow \frac{\mathrm{a}}{3}=\frac{\mathrm{b}}{4}=\frac{\mathrm{c}}{5}=\mathrm{k}\text{ (say), }\mathrm{k}\ne 0$
Now, a² + b² = c²
$\Rightarrow \mathrm{\Delta ABC} \mathrm{isrightangled}.$
$\begin{array}{l}\therefore {\sin }^2\mathrm{A}+{\sin }^2\mathrm{B}={\sin }^2\mathrm{C}\\ \Rightarrow {\sin }^2\mathrm{A}+{\sin }^2\mathrm{B}+{\sin }^2\mathrm{C}=2{\sin }^2\mathrm{C}=2{\sin }^2{90}^{\circ }=2\end{array}$