One of the roots of the given equation $\left| {\,\begin{array}{*{20}{c}}{x + a}&b&c\\b&{x + c}&a\\c&a&{x + b}\end{array}\,} \right| = 0$ is

If the system of equation $3x - 2y + z = 0$, $\lambda x - 14y + 15z = 0$, $x + 2y + 3z = 0$ have a non-trivial solution, then $\lambda = $

Let $\alpha $ and $\beta $ be the roots of the equation $x^2 + x + 1 = 0.$ Then for $y \ne 0$ in $R,$ $\left| {\begin{array}{*{20}{c}}
{y\, + \,1}&\alpha &\beta \\
\alpha &{y\, + \,\beta }&1\\
\beta &1&{y\, + \,\alpha }
\end{array}} \right|$ is equal to

For what value of $k$ to the following system of equations possess a non-trivial solution ?

$x + ky + 3z = 0$   ;    $3x + ky + 2z = 0$  ; $2x + 3y + 4z = 0$

The determinant $\,\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&2&3\\1&3&6\end{array}\,} \right|$ is not equal to

Consider the system of linear equations

$x+y+z=5, x+2 y+\lambda^2 z=9$

$x+3 y+\lambda z=\mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?