Let $m$ and $M$ be respectively the minimum and maximum values of

$\left|\begin{array}{ccc}\cos ^{2} x & 1+\sin ^{2} x & \sin 2 x \\ 1+\cos ^{2} x & \sin ^{2} x & \sin 2 x \\ \cos ^{2} x & \sin ^{2} x & 1+\sin 2 x\end{array}\right|$.

Then the ordered pair $( m , M )$ is equal to

 

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 value of $\left| {\,\begin{array}{*{20}{c}}{41}&{42}&{43}\\{44}&{45}&{46}\\{47}&{48}&{49}\end{array}\,} \right| = $

If $\omega $ is cube root of unity, then root of the equation $\left| {\begin{array}{*{20}{c}}
  {x + 2}&\omega &{{\omega ^2}} \\ 
  \omega &{x + 1 + {\omega ^2}}&1 \\ 
  {{\omega ^2}}&1&{x + 1 + \omega } 
\end{array}} \right| = 0$ is 

Show that points $A(a, b+c), B(b, c+a), C(c, a+b)$ are collinear

Let for any three distinct consecutive terms $a, b, c$ of an $A.P,$ the lines $a x+b y+c=0$ be concurrent at the point $\mathrm{P}$ and $\mathrm{Q}(\alpha, \beta)$ be a point such that the system of equations $ x+y+z=6, $ $ 2 x+5 y+\alpha z=\beta$ and $x+2 y+3 z=4$, has infinitely many solutions. Then $(P Q)^2$ is equal to________.