The total number of distinct x $$\in$$ R for which

$$\left| {\matrix{ x & {{x^2}} & {1 + {x^3}} \cr {2x} & {4{x^2}} & {1 + 8{x^3}} \cr {3x} & {9{x^2}} & {1 + 27{x^3}} \cr } } \right| = 10$$ is ______________.

Let $$z = {{ - 1 + \sqrt 3 i} \over 2}$$, where $$i = \sqrt { - 1} $$, and r, s $$\in$$ {1, 2, 3}. Let $$P = \left[ {\matrix{ {{{( - z)}^r}} & {{z^{2s}}} \cr {{z^{2s}}} & {{z^r}} \cr } } \right]$$ and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P² = $$-$$I is ____________.

Let M be a 3 $$\times$$ 3 matrix satisfying $$M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right]$$, $$M\left[ {\matrix{ 1 \cr { - 1} \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr { - 1} \cr } } \right]$$ and $$M\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr {12} \cr } } \right]$$. Then the sum of the diagonal entries of M is ___________.

Let $k$ be a positive real number and let

$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{array}{ccc} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{array}\right] . \end{aligned} $$

If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$

is equal to _________.

[ Note : adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$ ].