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

$\left| {\,\begin{array}{*{20}{c}}1&5&\pi \\{{{\log }_e}e}&5&{\sqrt 5 }\\{{{\log }_{10}}10}&5&e\end{array}\,} \right| = $

The system of equations ${x_1} - {x_2} + {x_3} = 2,$ $\,3{x_1} - {x_2} + 2{x_3} = - 6$ and $3{x_1} + {x_2} + {x_3} = - 18$ has

If $\left| {\begin{array}{*{20}{c}}
{a - b - c}&{2a}&{2a}\\
{2b}&{b - c - a}&{2b}\\
{2c}&{2c}&{c - a - b}
\end{array}} \right|$ $ = \left( {a + b + c} \right)\,{\left( {x + a + b + c} \right)^2}$ , $x   \ne 0$ and $a + b + c \ne 0$, then $x$ is equal to

If $\left| {\,\begin{array}{*{20}{c}}{{x^2} + x}&{x + 1}&{x - 2}\\{2{x^2} + 3x - 1}&{3x}&{3x - 3}\\{{x^2} + 2x + 3}&{2x - 1}&{2x - 1}\end{array}\,} \right| = Ax - 12$, then the value of $A $ is

The number of real values of $\lambda $ for which the system of linear equations $2x + 4y - \lambda  z = 0$ ;$4x + \lambda y + 2z = 0$ ; $\lambda x + 2y+ 2z = 0$ has infinitely many solutions, is