Question

If $a\ne b\ne c$ such that $\left|\begin{array}{ccc}{a}^3-1& b^3-1& c^3-1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|=0$ then

• A. $ab+bc+ca=0$
• B. $a+b+c=0$
• C. $abc=1$
• D. $a+b+c=1$

Answer 1

Answer: (C)

$abc=1$

$\left|\begin{array}{ccc}{a}^3-1& b^3-1& c^3-1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|=0$
$\left|\begin{array}{ccc}{a}^3& b^3& c^3\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|-1\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|=0$
$abc\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ 1& 1& 1\\ a& b& c\end{array}\right|-1\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|=0$
$abc\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ 1& 1& 1\\ a& b& c\end{array}\right|-1\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ 1& 1& 1\\ a& b& c\end{array}\right|=0$
$(abc-1)\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ 1& 1& 1\\ a& b& c\end{array}\right|=0$
$\Rightarrow abc-1=0$

$\Rightarrow abc=1$