Let the product of and be , . Let p and q be the maximum and the minimum values of respectively. Then p + q is equal to : [2025]

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Solution

Q.
Let the product of $ω_{1} = (8 + i) \sin θ + (7 + 4 i) \cos θ$ and $ω_{2} = (1 + 8 i) \sin θ + (4 + 7 i) \cos θ$ be $α + i β$ , $i = \sqrt{- 1}$ . Let p and q be the maximum and the minimum values of $α + β$ respectively. Then p + q is equal to : [2025]

1 140

2 130

3 160

4 150

Ans.
(2)

We have $ω_{1} = (8 \sin θ + 7 \cos θ) + i (\sin θ + 4 \cos θ)$ and $ω_{2} = (\sin θ + 4 \cos θ) + i (8 \sin θ + 7 \cos θ)$

Now,   $ω_{1} ω_{2} = 8 \sin^{2} θ + 39 \sin θ \cos θ + 28 \cos^{2} θ - 8 \sin^{2} θ - 39 \cos θ \sin θ - 28 \cos^{2} θ$

   $+ i (\sin^{2} θ + 16 \cos^{2} θ + 8 \sin θ \cos θ + 64 \sin^{2} θ + 49 \cos^{2} θ + 112 \sin θ \cos θ)$

Now, $α + i β = i (65 + 120 \sin θ \cos θ) = i (65 + 60 \sin 2 θ)$

(when sin 2 $θ$ = 1)

   $p = | α + β |_{m a x} = 65 + 60 = 125$

(when sin 2 $θ$ = –1)

$q = | α + β |_{\min} = 65 - 60 = 5$

$∴$ p + q = 125 + 5 =130.

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