If the system of equations $x + y+  z = 5$ ; $x + 2y + 3z = 9$ ; $x + 3y + \alpha z = \beta $ has infinitely many solutions, then $\beta  – \alpha $ equals

If $D = \left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 + x}&1\\1&1&{1 + y}\end{array}\,} \right|$ for $x \ne 0,y \ne 0$ then $D$ is

If ${A_\lambda } = \left( {\begin{array}{*{20}{c}}
\lambda &{\lambda  – 1}\\
{\lambda  – 1}&\lambda 
\end{array}} \right);\,\lambda  \in N$ then $|A_1| + |A_2| + ….. + |A_{300}|$ is equal to

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

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.