If $ 5$  is one root of the equation $\left| {\,\begin{array}{*{20}{c}}x&3&7\\2&x&{ - 2}\\7&8&x\end{array}\,} \right| = 0$, then other two roots of the equation are

For how many diff erent values of $a$ does the following system have at least two distinct solutions?

$a x+y=0$

$x+(a+10) y=0$

The following system of linear equations  $2 x+3 y+2 z=9$ ; $3 x+2 y+2 z=9$  ;$x-y+4 z=8$

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 $\omega $ is a cube root of unity and $\Delta = \left| {\begin{array}{*{20}{c}}1&{2\omega }\\\omega &{{\omega ^2}}\end{array}} \right|$, then ${\Delta ^2}$ is equal to

$\left| {\,\begin{array}{*{20}{c}}{{{\sin }^2}x}&{{{\cos }^2}x}&1\\{{{\cos }^2}x}&{{{\sin }^2}x}&1\\{ - 10}&{12}&2\end{array}\,} \right| = $