Let $\lambda, \mu \in R$. If the system of equations

$ 3 x+5 y+\lambda z=3 $

$ 7 x+11 y-9 z=2 $

$ 97 x+155 y-189 z=\mu$

has infinitely many solutions, then $\mu+2 \lambda$ is equal to :

$\left| {\begin{array}{*{20}{c}}0&a&{ - b}\\{ - a}&0&c\\b&{ - c}&0\end{array}} \right| = $

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

If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,\theta }&{\cos \,\theta } \\ 
  {\sin \,\theta }&{ - x}&1 \\ 
  {\cos \,\theta }&1&x 
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,2\theta }&{\cos \,\,2\theta } \\ 
  {\sin \,2\theta }&{ - x}&1 \\ 
  {\cos \,\,2\theta }&1&x 
\end{array}} \right|$, $x \ne 0$ ; then for all $\theta  \in \left( {0,\frac{\pi }{2}} \right)$

Let $a_1,a_2,a_3,....,a_{10}$ be in $G.P.$ with $a_i > 0$ for $i = 1, 2,....,10$ and $S$ be the set of pairs $(r,k), r, k \in N$ (the set of natural numbers) for which

$\left| {\begin{array}{*{20}{c}}
  {{{\log }_e}\,a_1^ra_2^k}&{{{\log }_e}\,a_2^ra_3^k}&{{{\log }_e}\,a_3^ra_4^k} \\
  {{{\log }_e}\,a_4^ra_5^k}&{{{\log }_e}\,a_5^ra_6^k}&{{{\log }_e}\,a_6^ra_7^k} \\ 
  {{{\log }_e}\,a_7^ra_8^k}&{{{\log }_e}\,a_8^ra_9^k}&{{{\log }_e}\,a_9^ra_{10}^k} 
\end{array}} \right| = 0$

Then the number of elements in $S$, is

The value of a for which the system of equations ${a^3}x + {(a + 1)^3}y + {(a + 2)^3}z = 0,$ $ax + (a + 1)y + (a + 2)z = 0,$ $x + y + z = 0,$ has a non zero solution is