Question

If in a $△ABC,si{n}^{3}A+si{n}^{3}B+si{n}^{3}C=3sinAsinBsinC$ then the value of the determinant
$\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ is

• A. 0
• B.
• C. (a+b+c)(ab+bc+ca)
• D. None of these

Answer 1

The correct option is A

0

$\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ = (a+b+c)$\left[\begin{array}{ccc}1& 1& 1\\ b& c& a\\ c& a& b\end{array}\right]$

=$(a+b+c)(bc+ca+ab-a^2-b^2-c^2)$
=-$(a^3+b^3+c^3-3abc)$
=-$8R^3(si{n}^{3}A+si{n}^{3}B+si{n}^{3}C-3sinAsinBsinC)=0$