The sum of the real roots of the equation $\left| {\begin{array}{*{20}{c}}
x&{ - 6}&{ - 1}\\
2&{ - 3x}&{x - 3}\\
{ - 3}&{2x}&{x = 2}
\end{array}} \right| = 0$ is equal to

Evaluate the determinants

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

$\left| {\,\begin{array}{*{20}{c}}{{{\sin }^2}x}&{{{\cos }^2}x}&1\\{{{\cos }^2}x}&{{{\sin }^2}x}&1\\{ - 10}&{12}&2\end{array}\,} \right| = $

Let $\mathrm{A}=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right],$ where $0 \leq \theta \leq 2 \pi$. Then

The value of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu$ has no solution, are :

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

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