For the first order reaction
2N₂O₅ (g) $$\to$$ 4NO₂ (g) + O₂ (g)

Oxidation states of the metal in the minerals haematite and magnetite, respectively are

Let $$\omega = {e^{{{i\pi } \over 3}}}$$, and a, b, c, x, y, z be non-zero complex numbers such that
$$a + b + c = x$$
$$a + b\omega + c{\omega ^2} = y$$
$$a + b{\omega ^2} + c\omega = z$$

Then the value of $${{{{\left| x \right|}^2} + {{\left| y \right|}^2} + {{\left| z \right|}^2}} \over {{{\left| a \right|}^2} + {{\left| b \right|}^2} + {{\left| c \right|}^2}}}$$ is

The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$