If $A = \int\limits_1^{\sin \theta } {\frac{t}{{1 + {t^2}}}} dt$ and $B = \int\limits_1^{\cos ec\theta } {\frac{dt}{{t\left( {1 + {t^2}} \right)}}} $ , (where $\theta  \in \left( {0,\frac{\pi }{2}} \right))$, then the-value of $\left| {\begin{array}{*{20}{c}}
A&{{A^2}}&{ - B}\\
{{e^{A + B}}}&{{B^2}}&{ - 1}\\
1&{{A^2} + {B^2}}&{ - 1}
\end{array}} \right|$ is

If $D(x) =$ $\left| {\begin{array}{*{20}{c}}{x - 1}&{{{(x - 1)}^2}}&{{x^3}}\\{x - 1}&{{x^2}}&{{{(x + 1)}^3}}\\x&{{{(x + 1)}^2}}&{{{(x + 1)}^3}}\end{array}} \right|$ then the coefficient of $x$ in $D(x)$ is

If $A = \left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right|,B = \left| {\,\begin{array}{*{20}{c}}1&1&1\\{{a^2}}&{{b^2}}&{{c^2}}\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right|,C = \left| {\,\begin{array}{*{20}{c}}a&b&c\\{{a^2}}&{{b^2}}&{{c^2}}\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right|,$ then which relation is correct

For what value of $\lambda $, the system of equations $x + y + z = 6,x + 2y + 3z = 10,$ $x + 2y + \lambda z = 12$ is inconsistent  $\lambda =$ ........

The following system of linear equations  $7 x+6 y-2 z=0$ ; $3 x+4 y+2 z=0$ ; ${x}-2{y}-6{z}=0,$ has

The system of linear equations  $3 x-2 y-k z=10$; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ is inconsistent if