If $\left| {\,\begin{array}{*{20}{c}}{x + 1}&1&1\\2&{x + 2}&2\\3&3&{x + 3}\end{array}\,} \right| = 0,$ then $x$ is

If the system of equations, $x + 2y -3z = 1, (k + 3) z = 3, (2k + 1)x + z = 0$ is inconsistent, then the value of $k$ is :-

If ${\Delta _1} = \left| {\,\begin{array}{*{20}{c}}x&b&b\\a&x&b\\a&a&x\end{array}\,} \right|$ and ${\Delta _2} = \left| {\,\begin{array}{*{20}{c}}x&b\\a&x\end{array}\,} \right|$ are the given determinants, then

Let $a, b, c > 0$ and $\Delta  = \left| \begin{gathered}
  a + b\,\,b\,\,c \hfill \\
  b\, + \,c\,\,c\,\,\,a \hfill \\
  c + a\,\,a\,\,b \hfill \\ 
\end{gathered}  \right| ,$ then which of the following is not correct?

$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = $

The determinant $\,\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&2&3\\1&3&6\end{array}\,} \right|$ is not equal to