If the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]$$ satisfies the equation

$${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & 0 \cr 0 & 0 & 1 \cr } } \right]$$ for some real numbers $$\alpha$$ and $$\beta$$, then $$\beta$$ $$-$$ $$\alpha$$ is equal to ___________.

Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha$$ + $$\beta$$ = 1 and $$\alpha$$$$\beta$$ = $$-$$1. Let p_n = ($$\alpha$$)ⁿ + ($$\beta$$)ⁿ, p_n$$-$$1 = 11 and p_n+1 = 29 for some integer n $$ \ge $$ 1. Then, the value of p$$_n^2$$ is ___________.

Let z be those complex numbers which satisfy

| z + 5 | $$ \le $$ 4 and z(1 + i) + $$\overline z $$(1 $$-$$ i) $$ \ge $$ $$-$$10, i = $$\sqrt { - 1} $$.

If the maximum value of | z + 1 |² is $$\alpha$$ + $$\beta$$$$\sqrt 2 $$, then the value of ($$\alpha$$ + $$\beta$$) is ____________.

If the arithmetic mean and geometric mean of the p^th and q^th terms of the
sequence $$-$$16, 8, $$-$$4, 2, ...... satisfy the equation
4x² $$-$$ 9x + 5 = 0, then p + q is equal to __________.