Let $A =$ $\left[ {\begin{array}{*{20}{c}}1&{\sin \theta }&1\\{ - \sin \theta }&1&{\sin \theta }\\{ - 1}&{ - \sin \theta }&1\end{array}} \right]$, where $0 \le \theta < 2\pi$ , then

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

The determinant $\left| {\,\begin{array}{*{20}{c}}a&b&{a - b}\\b&c&{b - c}\\2&1&0\end{array}\,} \right|$ is equal to zero if $a,b,c$ are in

If $\alpha , \beta \, and \, \gamma$ are real numbers , then $D = \left|{\begin{array}{*{20}{c}}1&{\cos \,(\beta \, - \,\alpha )}&{\cos \,(\gamma \, - \,\alpha )}\\{\cos \,(\alpha \, - \,\beta )}&1&{\cos \,(\gamma \, - \,\beta )}\\{\cos \,(\alpha \, - \,\gamma )}&{\cos \,(\beta \, - \,\gamma )}&1 \end{array}} \right|$ =

$\left| {\,\begin{array}{*{20}{c}}{a - 1}&a&{bc}\\{b - 1}&b&{ca}\\{c - 1}&c&{ab}\end{array}\,} \right| = $

If $\left| {\,\begin{array}{*{20}{c}}{x + 1}&3&5\\2&{x + 2}&5\\2&3&{x + 4}\end{array}\,} \right| = 0$, then $ x =$