If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x³) + x g(x³) is divisible by x² + x + 1, then P(1) is equal to ___________.

The term independent of x in the expansion of

$${\left[ {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right]^{10}}$$, x $$\ne$$ 1, is equal to ____________.

Let y = y(x) be the solution of the differential equation

xdy $$-$$ ydx = $$\sqrt {({x^2} - {y^2})} dx$$, x $$ \ge $$ 1, with y(1) = 0. If the area bounded by the line x = 1, x = e^$$\pi$$, y = 0 and y = y(x) is $$\alpha$$e^2$$\pi$$ + $$\beta$$, then the value of 10($$\alpha$$ + $$\beta$$) is equal to __________.

Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3} \cr } } \right]$$. Then the value of n$$\in$$N for which Pⁿ = 5I $$-$$ 8P is equal to ____________.