Let the product of $\omega_1=(8+i) \sin heta+(7+4 i) \cos heta$ and $\omega_2=(1+8 i) \sin heta+(4+7 i) \cos heta$ be $\alpha+i \beta$, $\mathrm{i}=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha+\beta$ respectively.

Question

Let the product of $\omega_1=(8+i) \sin 	heta+(7+4 i) \cos 	heta$ and $\omega_2=(1+8 i) \sin 	heta+(4+7 i) \cos 	heta$ be $\alpha+i \beta$, $\mathrm{i}=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha+\beta$ respectively.

Aicharya · Accepted Answer

$\begin{aligned} & \omega_1=(8 \sin 	heta+7 \cos 	heta)+i(\sin 	heta+4 \cos 	heta) \ & \omega_2=(\sin 	heta+4 \cos 	heta)+i(8 \sin 	heta+7 \cos 	heta) \ & \omega_1 \omega_2=8 \sin ^2 	heta+7 \sin 	heta \cos 	heta+32 \sin 	heta \cos 	heta+ \ & 28 \cos ^2 	heta-8 \sin ^2 	heta-32 \sin 	heta \cos 	heta-7 \sin 	heta \cos 	heta \ & -28 \cos ^2 	heta+i\left(\sin ^2 	heta+4 \sin 	heta \cos 	heta+4 \sin 	heta \cos 	hetaight. \ & +16 \cos ^2 	heta+64 \sin ^2 	heta+56 \sin 	heta \cos 	heta+56 \sin 	heta \ & \left.\cos 	heta+49 \cos ^2 	hetaight) \ & \omega_1 \omega_2=0+i\left(65 \sin ^2 	heta+120 \sin 	heta \cos 	heta+65 \cos ^2 	hetaight) \ & \alpha+\beta=65+60 \sin 2 q \ & \alpha+\left.\betaight|_{\max }=125 \ & \alpha+\left.\betaight|_{\min }=5 \end{aligned}$ $	ext { Ans. }=125+5=130$ option (2)

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i) \sin \theta+(4+7…

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