If $f(\theta ) =\left| {\begin{array}{*{20}{c}}
1&{\cos {\mkern 1mu} \theta }&1\\
{ - \sin {\mkern 1mu} \theta }&1&{ - \cos {\mkern 1mu} \theta }\\
{ - 1}&{\sin {\mkern 1mu} \theta }&1
\end{array}} \right|$ and $A$ and $B$ are respectively the maximum and the minimum values of $f(\theta )$, then $(A , B)$ is equal to

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 $A\, = \,\left[ \begin{gathered}
  1\ \ \ \,1\ \ \ \,2\ \ \  \hfill \\
  0\ \ \ \,2\ \ \ \,1\ \ \  \hfill \\
  1\ \ \ \,0\ \ \ \,2\ \ \  \hfill \\ 
\end{gathered}  \right]$ 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

Statement $-1$ : The system of linear equations

$x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0$

$x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0$

$x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0$

has a non-trivial solution for only one value of $\alpha $ lying in the interval $\left( {0\,,\,\frac{\pi }{2}} \right)$ 

Statement $-2$ : The equation in $\alpha $

$\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0$

has only one solution lying in the interval $\left( {0\,,\,\frac{\pi }{2}} \right)$

Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations

$\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$

$\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$

$\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$

has a non-trivial solution, then the value of $\theta$ is :