Let P = $$\left[ {\matrix{ 3 & { - 1} & { - 2} \cr 2 & 0 & \alpha \cr 3 & { - 5} & 0 \cr } } \right]$$, where $$\alpha $$ $$ \in $$ R. Suppose Q = [ q_ij] is a matrix satisfying PQ = kl₃ for some non-zero k $$ \in $$ R.
If q₂₃ = $$ - {k \over 8}$$ and |Q| = $${{{k^2}} \over 2}$$, then a² + k² is equal to ______.

The minimum value of $$\alpha $$ for which the
equation $${4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha $$ has at least one solution in $$\left( {0,{\pi \over 2}} \right)$$ is .......

If the least and the largest real values of a, for which the
equation z + $$\alpha $$|z – 1| + 2i = 0 (z $$ \in $$ C and i = $$\sqrt { - 1} $$) has a solution, are p and q respectively; then 4(p² + q²) is equal to __________.

Let three vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow c $$ is coplanar
with $$\overrightarrow a $$ and $$\overrightarrow b $$, $$\overrightarrow a .\overrightarrow c $$ = 7 and $$\overrightarrow b $$ is perpendicular to $$\overrightarrow c $$, where
$$\overrightarrow a = - \widehat i + \widehat j + \widehat k$$ and $$\overrightarrow b = 2\widehat i + \widehat k$$ , then the
value of $$2{\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2}$$ is _____.