Let and respectively be the maximum and the minimum values of the function Then is equal to: [2026]

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Solution

Q.
Let $α$ and $β$ respectively be the maximum and the minimum values of the function

$f (θ) = 4 (\sin^{4} (\frac{7 π}{2} - θ) + \sin^{4} (11 π + θ)) - 2 (\sin^{6} (\frac{3 π}{2} - θ) + \sin^{6} (9 π - θ)), θ \in R .$

Then $α + 2 β$ is equal to: [2026]

1 5

2 3

3 4

4 6

Ans.
(1)

$f (θ) = 4 (\sin^{4} (\frac{7 π}{2} - θ) + \sin^{4} (11 π + θ)) - 2 (\sin^{6} (\frac{3 π}{2} - θ) + \sin^{6} (9 π - θ))$

$f (θ) = 4 (\cos^{4} θ + \sin^{4} θ) - 2 (\cos^{6} θ + \sin^{6} θ)$

$f (θ) = 4 (1 - 2 \sin^{2} θ \cos^{2} θ) - 2 (1 - 3 \sin^{2} θ \cos^{2} θ)$

$f (θ) = 2 - 2 \sin^{2} θ \cos^{2} θ$

$f (θ) = 2 - \frac{\sin^{2} (2 θ)}{2}$

$α = f {(θ)}_{\max} = 2$

$β = f {(θ)}_{\min} = \frac{3}{2}$

$\Rightarrow α + 2 β = 5$

$Ans. = 5 option (1)$

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