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

If $\left| {\,\begin{array}{*{20}{c}}{1 + ax}&{1 + bx}&{1 + cx}\\{1 + {a_1}x}&{1 + {b_1}x}&{1 + {c_1}x}\\{1 + {a_2}x}&{1 + {b_2}x}&{1 + {c_2}x}\end{array}\,} \right|,$ $ = {A_0} + {A_1}x + {A_2}{x^2} + {A_3}{x^3}$ then ${A_1}$ is equal to

Consider the system of linear equations ${a_1}x + {b_1}y + {c_1}z + {d_1} = 0$, ${a_2}x + {b_2}y + {c_2}z + {d_2} = 0$ and ${a_3}x + {b_3}y + {c_3}z + {d_3} = 0$. Let us denote by $\Delta (a,b,c)$ the determinant $\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ if $\Delta (a,b,c) \ne 0$, then the value of $x$ in the unique solution of the above equations is

Let $\omega $ be a complex number such that  $2\omega + 1 = z$ where $z = \sqrt { – 3} $ . If $\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ – {\omega ^2} – 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ then $k$ is equal to :

$\left| {\,\begin{array}{*{20}{c}}{{{({a^x} + {a^{ – x}})}^2}}&{{{({a^x} – {a^{ – x}})}^2}}&1\\{{{({b^x} + {b^{ – x}})}^2}}&{{{({b^x} – {b^{ – x}})}^2}}&1\\{{{({c^x} + {c^{ – x}})}^2}}&{{{({c^x} – {c^{ – x}})}^2}}&1\end{array}\,} \right| = $