If a rectangle is inscribed in an equilateral triangle of side length $$2\sqrt 2 $$ as shown in the figure, then the square of the largest area of such a rectangle is _____________.

If $$\left( {\overrightarrow a + 3\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 5\overrightarrow b } \right)$$ and $$\left( {\overrightarrow a - 4\overrightarrow b } \right)$$ is perpendicular to $$\left( {7\overrightarrow a - 2\overrightarrow b } \right)$$, then the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ (in degrees) is _______________.

Let a curve y = f(x) pass through the point (2, (log_e2)²) and have slope $${{2y} \over {x{{\log }_e}x}}$$ for all positive real value of x. Then the value of f(e) is equal to ______________.

If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a⁴ + b⁴ + c⁴ is equal to ______________.