Exams Joint Entrance Examination Mathematics Complex Numbers Question 455 Question 455 of 455 Medium MCQ
Let the product of ω 1 = ( 8 + i ) sin θ + ( 7 + 4 i ) cos θ \omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta ω 1 = ( 8 + i ) sin θ + ( 7 + 4 i ) cos θ and ω 2 = ( 1 + 8 i ) sin θ + ( 4 + 7 i ) cos θ \omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta ω 2 = ( 1 + 8 i ) sin θ + ( 4 + 7 i ) cos θ be α + i β \alpha+i \beta α + i β , i = − 1 i=\sqrt{-1} i = − 1 . Let p and q be the maximum and the minimum values of α + β \alpha+\beta α + β respectively. Then p + q \mathrm{p}+\mathrm{q} p + q is equal to :
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\[ ω 1 = ( 8 sin θ + 7 cos θ ) + i ( sin θ + 4 cos θ ) ω 2 = ( sin θ + 4 cos θ ) + i ( 8 sin θ + 7 cos θ ) α = ( 8 sin θ + 7 cos θ ) + ( sin θ + 4 cos θ ) − ( sin θ + 4 cos θ ) + ( 8 sin θ + 7 cos θ ) = 0 β = ( 8 sin θ + 7 cos θ ) 2 + ( sin θ + 4 cos θ ) 2 \begin{aligned}
& \omega_1=(8 \sin \theta+7 \cos \theta)+i(\sin \theta+4 \cos \theta) \\
& \omega_2=(\sin \theta+4 \cos \theta)+i(8 \sin \theta+7 \cos \theta) \\
& \alpha=(8 \sin \theta+7 \cos \theta)+(\sin \theta+4 \cos \theta) \\
& \quad-(\sin \theta+4 \cos \theta)+(8 \sin \theta+7 \cos \theta)=0 \\
& \beta=(8 \sin \theta+7 \cos \theta)^2+(\sin \theta+4 \cos \theta)^2
\end{aligned} ω 1 = ( 8 sin θ + 7 cos θ ) + i ( sin θ + 4 cos θ ) ω 2 = ( sin θ + 4 cos θ ) + i ( 8 sin θ + 7 cos θ ) α = ( 8 sin θ + 7 cos θ ) + ( sin θ + 4 cos θ ) − ( sin θ + 4 cos θ ) + ( 8 sin θ + 7 cos θ ) = 0 β = ( 8 sin θ + 7 cos θ ) 2 + ( sin θ + 4 cos θ ) 2 \]
\[ = 65 sin 2 θ + 65 cos 2 θ + 56 sin 2 θ + 4 sin 2 θ = 65 + 60 sin 2 θ ( α + β ) max = 125 = p ( α + β ) min = 5 = q p + q = 130 \begin{aligned}
& =65 \sin ^2 \theta+65 \cos ^2 \theta+56 \sin 2 \theta+4 \sin 2 \theta \\
& =65+60 \sin 2 \theta \\
& (\alpha+\beta)_{\max }=125=p \\
& (\alpha+\beta)_{\min }=5=q \\
& p+q=130
\end{aligned} = 65 sin 2 θ + 65 cos 2 θ + 56 sin 2 θ + 4 sin 2 θ = 65 + 60 sin 2 θ ( α + β ) m a x = 125 = p ( α + β ) m i n = 5 = q p + q = 130 \] Answer the question to unlock