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 all are different and non zero), then value of $a+b+c$ is

• A. 0
• B. 0.0
• C. 0.00

Answer 1

Answer: (A), (B), (C)

$\Delta =0$
$\Rightarrow bc\cdot ca\cdot ab\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$

$\Rightarrow -a^2{b}^2{c}^2\cdot (a+b+c)\cdot \frac{1}{2}\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]=0$

$\Rightarrow a+b+c=0$