Question

In $\triangle A B C$, if $\left|\begin{array}{lll}1 & a & b \\ 1 & c & a \\ 1 & b & c\end{array}\right|=0$, then $\sin ^2 A+\sin ^2 B+\sin ^2 C$ is equal to

(A) $\frac{4}{9}$

(B) $\frac{9}{4}$

(C) $3 \sqrt{3}$

(D) 1

Answer 1

Given determinant is satisfied, if $a=b=c$

$\begin{aligned}\Rightarrow \quad A=B=C & =60^{\circ} \\\therefore \sin ^2 A+\sin ^2 B+\sin ^2 C & =3 \sin ^2\left(60^{\circ}\right) \\& =3 \times\left(\frac{\sqrt{3}}{2}\right)^2=\frac{9}{4}\end{aligned}$

So, The correct option is (B).