Let the product of _1=(8+i) +(7+4 i) and _2=(1+8 i ) +(4+7 i ) be… - Vidyadip

MCQ

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

A
140
✓
130
C
160
D
150

✓

Answer

Correct option: B.

130

(B) 130
$\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) $
$\omega_1 \omega_2=8 \sin ^2 \theta+7 \sin \theta \cos \theta+32 \sin \theta \cos \theta+$
$28 \cos ^2 \theta-8 \sin ^2 \theta-32 \sin \theta \cos \theta-7 \sin \theta \cos \theta$
$-28 \cos ^2 \theta+i\left(\sin ^2 \theta+4 \sin \theta \cos \theta+4 \sin \theta \cos \theta\right.$
$+16 \cos ^2 \theta+64 \sin ^2 \theta+56 \sin \theta \cos \theta+56 \sin \theta \left.\cos \theta+49 \cos ^2 \theta\right) $
$\omega_1 \omega_2=0+i\left(65 \sin ^2 \theta+120 \sin \theta \cos \theta+65 \cos ^2 \theta\right) $
$\alpha+\beta=65+60 \sin 2 q$
$\alpha+\left.\beta\right|_{\max }=125 $
$\alpha+\left.\beta\right|_{\min }=5 $
$\text { Ans. }=125+5=130$

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