Question

If $a\in R-\left\{0\right\}$ and $\left|\begin{array}{ccc}x+1& x& x\\ x& x+a& x\\ x& x& x+a^2\end{array}\right|=0,$ then $x$ belongs to

• A. $R^{-}$
• B. $R^{+}$
• C. $Z$
• D. $N$

Answer 1

The correct option is A $R^{-}$

$\left|\begin{array}{ccc}x+1& x& x\\ x& x+a& x\\ x& x& x+a^2\end{array}\right|=0$
Using the transformation, $C_3\to C_3-C_1,C_2\to C_2-C_1$
$\Rightarrow \left|\begin{array}{ccc}x+1& -1& -1\\ x& a& 0\\ x& 0& a^2\end{array}\right|=0$
$\Rightarrow (a^3+a^2+a)x=-a^3$
$\Rightarrow x=\frac{-a^3}{a^3+a^2+a}$
$\Rightarrow x=\frac{-a^2}{a^2+a+1}<0$