Given, $$\alpha$$ is a root of the equation 1 + x² + x⁴ = 0

$$\therefore$$ $$\alpha$$ will satisfy the equation.

$$\therefore$$ 1 + $$\alpha$$² + $$\alpha$$⁴ = 0

$${\alpha ^2} = {{ - 1 \pm \sqrt {1 - 4} } \over 2}$$

$$ = {{ - 1 \pm \sqrt 3 i} \over 2}$$

$$\therefore$$ $${\alpha ^2} = \omega \,ar\,{\omega ^2}$$

Now,

$${\alpha ^{1011}} + {\alpha ^{2022}} - {\alpha ^{3033}}$$

$$ = \alpha \,.\,{({\alpha ^2})^{505}} + {({\alpha ^2})^{1011}} - \alpha \,.\,{({\alpha ^2})^{1516}}$$

$$ = \alpha {(\omega )^{505}} + {(\omega )^{1011}} - \alpha \,.\,{(\omega )^{1516}}$$

$$ = \alpha \,.\,{({\omega ^3})^{168}}\,.\,\omega + {({\omega ^3})^{337}} - \alpha \,.\,{({\omega ^3})^{505}}\,.\,\omega $$

$$ = \alpha \,\omega + 1 - \alpha \,\omega $$

$$ = 1$$