Let and be the distinct roots of . If m and M are the minimum and the maximum values of , then 16(M + m) equals: [2025]

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Solution

Q.
Let $α_{θ}$ and $β θ$ be the distinct roots of $2 x^{2} + (\cos θ) x - 1 = 0, θ \in (0, 2 π)$ . If m and M are the minimum and the maximum values of $α_{θ}^{4} + β_{θ}^{4}$ , then 16(M + m) equals: [2025]

1 24

2 25

3 17

4 27

Ans.
(2)

$α_{θ}^{4} + β_{θ}^{4} = {(α_{θ}^{2} + β_{θ}^{2})}^{2} - 2 α_{θ}^{2} β_{θ}^{2}$

$= {[{(α_{θ} + β_{θ})}^{2} - 2 α_{θ} β_{θ}]}^{2} - 2 {(α_{θ} β_{θ})}^{2}$

$= {[\frac{\cos^{2} θ}{4} + 1]}^{2} - 2 (\frac{1}{4}) [∵ α_{θ} + β_{θ} = \frac{- \cos θ}{2} and α_{θ} β_{θ} = - \frac{1}{2}]$

$= {(\frac{\cos^{2} θ}{4} + 1)}^{2} - \frac{1}{2}$

Since, $0 \leq \cos^{2} θ \leq 1$

$∴ M = \frac{25}{16} - \frac{1}{2} = \frac{17}{16}$ and $m = \frac{1}{2}$ .

Hence, 16(M + m) = 25

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