If $a_i^2 + b_i^2 + c_i^2 = 1,\,i = 1,2,3$ and $a_ia_j + b_ib_j +c_ic_j = 0$ $\left( {i \ne j,i,j = 1,2,3} \right)$ then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\ 
  {{b_1}}&{{b_2}}&{{b_3}} \\ 
  {{c_1}}&{{c_2}}&{{c_3}} 
\end{array}} \right|$ is

If $A, B, C$ are the angles of triangle then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {\sin \,2A}&{\sin \,C}&{\sin \,B} \\ 
  {\sin \,C}&{\sin \,2B}&{\sin A} \\ 
  {\sin \,B}&{\sin \,A}&{\sin \,2C} 
\end{array}} \right|$ is

If the sum of squares of all real values of $\alpha$, for which the lines $2 x-y+3=0,6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $\mathrm{p}$ is $………$

If ${D_p} = \left| {\,\begin{array}{*{20}{c}}p&{15}&8\\{{p^2}}&{35}&9\\{{p^3}}&{25}&{10}\end{array}\,} \right|$, then ${D_1} + {D_2} + {D_3} + {D_4} + {D_5} = $

Let $S$ be the set of all values of $\theta \in[-\pi, \pi]$ for which the system of linear equations

$x+y+\sqrt{3} z=0$

$-x+(\tan \theta) y+\sqrt{7} z=0$

$x+y+(\tan \theta) z=0$

has non-trivial solution. Then $\frac{120}{\pi} \sum_{\theta \in s} \theta$ is equal to