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 ___________.

The sum of 162^th power of the roots of the equation x³ $$-$$ 2x² + 2x $$-$$ 1 = 0 is ________.

The number of the real roots of the equation $${(x + 1)^2} + |x - 5| = {{27} \over 4}$$ is ________.

The least positive value of 'a' for which the equation

2x² + (a – 10)x + $${{33} \over 2}$$ = 2a has real roots is