If $\left| {{\kern 1pt} \begin{array}{*{20}{c}}1&2&3\\2&x&3\\3&4&5\end{array}\,} \right| = 0,$ then  $x =$

How many values of $k $ , systeam of linear equations $\left( {k + 1} \right)x + 8y = 4k\;,\;kx + \left( {k + 3} \right)y$$ = 3k - 1$ has no solutions.

The remainder when the determinant $\left|\begin{array}{lll} 2014^{2014} & 2015^{2015} & 2016^{2016} \\ 2017^{2017} & 2018^{2018} & 2019^{2019} \\ 2020^{2020} & 2021^{2021} & 2022^{2022} \end{array}\right|$  is divided by $5$ is

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{{\omega ^2}}&\omega \\1&\omega &{{\omega ^2}}\end{array}\,} \right| = $

If $\alpha+\beta+\gamma=2 \pi$, then the system of equations

$x+(\cos \gamma) y+(\cos \beta) z=0$

$(\cos \gamma) x+y+(\cos \alpha) z=0$

$(\cos \beta) x+(\cos \alpha) y+z=0$

has :

Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3 x+2 y+5 z=3$ $9 x+4 y+(28+[\lambda]) z=[\lambda]$ has a solution is: