Mathematics - Complex Number Question with Solution | TestHub

$\begin{array}{rl}& \alpha +\beta =\mathrm{a}\alpha \beta =-\mathrm{b}\\ & {\mathrm{P}}_6={\mathrm{aP}}_5+{\mathrm{bP}}_4\\ & 45\sqrt{7}\mathrm{i}=\mathrm{a}\times 11\sqrt{7}\mathrm{i}+\mathrm{b}(-3\sqrt{7})\mathrm{i}\\ & 45=11\mathrm{a}-3\mathrm{b}\end{array}$ and $\begin{array}{rl}& P_5={\mathrm{aP}}_4+b{P}_3\\ & 11\sqrt{7}\mathrm{i}=\mathrm{a}(-3\sqrt{7}\mathrm{i})+\mathrm{b}(-5\sqrt{7}\mathrm{i})\\ & 11=-3\mathrm{a}-5\mathrm{b}\\ & \mathrm{a}=3,\mathrm{b}=-4\\ & \left|{\alpha }^4+{\beta }^4\right|=\sqrt{{\left({\alpha }^4-{\beta }^4\right)}^2+4{\alpha }^4{\beta }^4}\\ & =\sqrt{-63+4.4^4}\\ & =\sqrt{-63+1024}=\sqrt{961}=31\end{array}$