If $a \ne b \ne c,$ the value of $x$ which satisfies the equation $\left| {\,\begin{array}{*{20}{c}}0&{x - a}&{x - b}\\{x + a}&0&{x - c}\\{x + b}&{x + c}&0\end{array}\,} \right| = 0$, is

The system of equations : $2x\, \cos^2\theta + y\, \sin2\theta - 2\sin\theta = 0$ $x\, \sin2\theta + 2y\, \sin^2\theta = - 2\, \cos\theta$ $x\, \sin\theta - y \cos\theta = 0$ , for all values of $\theta$ , can

If ${\Delta _r} = \left| {\begin{array}{*{20}{c}}
  r&{2r - 1}&{3r - 2} \\ 
  {\frac{n}{2}}&{n - 1}&a \\ 
  {\frac{1}{2}n\left( {n - 1} \right)}&{{{\left( {n - 1} \right)}^2}}&{\frac{1}{2}\left( {n - 1} \right)\left( {3n - 4} \right)} 
\end{array}} \right|$ then the value of $\sum\limits_{r = 1}^{n - 1} {{\Delta _r}} $

For $\alpha, \beta \in R$, suppose the system of linear equations $x-y+z=5$ ; $ 2 x+2 y+\alpha z=8 $ ; $3 x-y+4 z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

If the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:

The solutions of the equation $\left| {\,\begin{array}{*{20}{c}}x&2&{ - 1}\\2&5&x\\{ - 1}&2&x\end{array}\,} \right| = 0$ are