Let k be a positive real number and  $A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right] , B=\left[\begin{array}{ccc}0& 2k-1& \sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$
If  $\det \left(adjA\right)+\det \left(adjB\right)={10}^6$ , then [k/2] is equal to ( where [.] is G.I.F) _______

det  $A=(2k-1)(4k^2-1)+2\sqrt{k}(4k\sqrt{k}+2\sqrt{k})+2\sqrt{k}(4k\sqrt{k}+2\sqrt{k})$
 $$\begin{gathered} =(2k-1)(4k^2-1)+8(2k+1)k \\ =(2k+1)({(2k-1)}^2+8k)={(2k+1)}^3 \end{gathered}$$
$\therefore \det B=0$  since B is skew symmetric matrix of order 3.
$\therefore \det \left(adjB\right)={\left(\det B\right)}^2=0$ 
 $\therefore {(2k+1)}^6={10}^6$
 $\Rightarrow 2k+1=10\Rightarrow k=4.5$
So, $[k/2]=[4.5/2]=2$