Mathematics - Complex Number Question with Solution | TestHub

$\begin{array}{rl}& {\omega }_1=(8\sin \theta +7\cos \theta )+i(\sin \theta +4\cos \theta )\\ & {\omega }_2=(\sin \theta +4\cos \theta )+i(8\sin \theta +7\cos \theta )\\ & {\omega }_1{\omega }_2=8{\sin }^2\theta +7\sin \theta \cos \theta +32\sin \theta \cos \theta +\\ & 28{\cos }^2\theta -8{\sin }^2\theta -32\sin \theta \cos \theta -7\sin \theta \cos \theta \\ & -28{\cos }^2\theta +i({\sin }^2\theta +4\sin \theta \cos \theta +4\sin \theta \cos \theta \\ & +16{\cos }^2\theta +64{\sin }^2\theta +56\sin \theta \cos \theta +56\sin \theta \\ & \cos \theta +49{\cos }^2\theta )\\ & {\omega }_1{\omega }_2=0+i\left(65{\sin }^2\theta +120\sin \theta \cos \theta +65{\cos }^2\theta \right)\\ & \alpha +\beta =65+60\sin 2q\\ & \alpha +{\beta |}_{\max }=125\\ & \alpha +{\beta |}_{\min }=5\end{array}$
$\text{ Ans. }=125+5=130$
option (2)