Question

If $\Delta =\left|\begin{array}{ccc}abc& b^{2}c& c^{2}b\\ abc& c^{2}a& c{a}^2\\ abc& a^{2}b& b^{2}a\end{array}\right|=0$ $(a,b,c\in R$
and are all different and non-zero), then the value of $a+b+c$ is:

• A. $3$
• B. $-1$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

Taking $bc,ac$ and $ab$ common from $R_1,R_2$ and $R_3$ respectively, we have
$\Delta =bc.ca.ab\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$
Expanding determinant along $R_1,$ we get
$\Delta =a^2{b}^2{c}^2(-a^3-b^3-c^3+3abc)$
$\Rightarrow -a^2{b}^2{c}^2(a^3+b^3+c^3-3abc)=0$
$\Rightarrow -a^2{b}^2{c}^2(a+b+c)\frac{1}{2}\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]=0$
Hence we have, $a+b+c=0$