Find values of ${x},$ if  $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$

The value of $\left| {\,\begin{array}{*{20}{c}}{{1^2}}&{{2^2}}&{{3^2}}\\{{2^2}}&{{3^2}}&{{4^2}}\\{{3^2}}&{{4^2}}&{{5^2}}\end{array}\,} \right|$ is

$\left| {\begin{array}{*{20}{c}}
{4 + {x^2}}&{ - 6}&{ - 2}\\
{ - 6}&{9 + {x^2}}&3\\
{ - 2}&3&{1 + {x^2}}
\end{array}} \right|$ $;(x\neq0)$ is not divisible by

$S$ denote the set of all real values of $\lambda$ such that the system of equations  $\lambda x + y + z =1$ ; $x +\lambda y + z =1$ ; $x + y +\lambda z =1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to

Evaluate the determinants

$\left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$

The number of real values of $\lambda $ for which the system of linear equations $2x + 4y - \lambda  z = 0$ ;$4x + \lambda y + 2z = 0$ ; $\lambda x + 2y+ 2z = 0$ has infinitely many solutions, is