Explanation

(d)
Discriminant of the equation $3x^{2} + 8x + 15 = 0$ is given by
$D = 64 - 180 = - 116 < 0$
So, its roots are imaginary and therefore roots are conjugate to each other. Therefore, one common root means both the roots are common.
$\therefore\ \frac{a}{3} = \frac{2b}{8} = \frac{3c}{15}$
$\Rightarrow \ \frac{\ a}{3} = \frac{b}{4} = \frac{c}{5} = k$ (say), $k \neq 0$
$\Rightarrow \ a = 3k,\ b = 4k,\ c = 5k$
Now, $a^{2} + b^{2} = c^{2}$
$\Rightarrow \ \mathrm{\Delta}ABC$ is right angled.
$\therefore{\ \sin^{2}}{A + \sin^{2}{B = \sin^{2}C}}$
$\Rightarrow {\ sin}^{2}{A + \sin^{2}{B + \sin^{2}{C = 2\sin^{2}C = 2\sin^{2}{90{^\circ} = 2}}}}$