The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is

If $\omega$ is one of the imaginary cube roots of unity, then the value of the determinant $\left| {\begin{array}{*{20}{c}}1&{{\omega ^3}}&{{\omega ^2}}\\ {{\omega ^3}}&1&\omega \\{{\omega ^2}}&\omega &1\end{array}} \right|$ $=$

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

Let $[.]$ , $ \{.\} $ and $sgn$$(.)$ denotes greatest integer function, fractional part function and signum function respectively, then value of determinant

$\left| {\begin{array}{*{20}{c}}
  {\left[ \pi  \right]}&{amp(1 + i\sqrt 3 )}&1 \\ 
  1&0&2 \\ 
  {\operatorname{sgn} ({{\cot }^{ - 1}}x)}&1&{\{ \pi \} } 
\end{array}} \right|$ is-

If $a,b,c$ are respectively the ${p^{th}},{q^{th}}{r^{th}}$terms of an $A.P.,$ the $\left| {\,\begin{array}{*{20}{c}}a&p&1\\b&q&1\\c&r&1\end{array}\,} \right| = $

If the system of equations  $2 x+3 y-z=5$  ;  $x+\alpha y+3 z=-4$  ;  $3 x-y+\beta z=7$ has infinitely many solutions, then $13 \alpha \beta$ is equal to