For how many diff erent values of $a$ does the following system have at least two distinct solutions?

$a x+y=0$

$x+(a+10) y=0$

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

If $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = k(a + b + c)({a^2} + {b^2} + {c^2}$ $ – bc – ca – ab)$, then $k =$

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

Let $\mathrm{A}=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right],$ where $0 \leq \theta \leq 2 \pi$. Then