If $A\, = \,\left[ \begin{gathered}
  1\ \ \ \,1\ \ \ \,2\ \ \  \hfill \\
  0\ \ \ \,2\ \ \ \,1\ \ \  \hfill \\
  1\ \ \ \,0\ \ \ \,2\ \ \  \hfill \\ 
\end{gathered}  \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

If $-9 $ is a root of the equation $\left| {\,\begin{array}{*{20}{c}}x&3&7\\2&x&2\\7&6&x\end{array}\,} \right| = 0$ then the other two roots are

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 $\theta  \in \left( {0,\frac{\pi }{2}} \right)$

The number of $\theta \in(0,4 \pi)$ for which the system of linear equations

$3(\sin 3 \theta) x-y+z=2$, $3(\cos 2 \theta) x+4 y+3 z=3$, $6 x+7 y+7 z=9$ has no solution is.

If the system of linear equations $2 \mathrm{x}+2 \mathrm{ay}+\mathrm{az}=0$ ; $2 x+3 b y+b z=0$ ; $2 \mathrm{x}+4 \mathrm{cy}+\mathrm{cz}=0$ ; where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then