If the range of f = sin 4 + 3 cos 2 sin 4 + cos 2 , R is, then the sum of the infinite G.P., whose first term is 64 and the common ratio is , is equal to _______. [2024]

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Solution

Q.
If the range of $f (θ) = \frac{\sin^{4} θ + 3 \cos^{2} θ}{\sin^{4} θ + \cos^{2} θ}, θ \in R$ is $[α, β],$ then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{α}{β},$ is equal to _______. [2024]

Ans.
(96)

We have, $f (θ) = \frac{\sin^{4} θ + 3 \cos^{2} θ}{\sin^{4} θ + \cos^{2} θ}$

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

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

Now, $\cos^{2} θ + \frac{1}{\cos^{2} θ} \geq 2$ $[∵ A . M . \geq G . M .]$

$\Rightarrow \frac{1}{\cos^{2} θ} + \cos^{2} θ - 1 \geq 1 \Rightarrow \cos^{2} θ + \frac{1}{\cos^{2} θ} - 1 \in [1, \infty)$

$\Rightarrow \frac{1}{\cos^{2} θ + \frac{1}{\cos^{2} θ} - 1} \in (0, 1]$

When $\cos θ = 0, f (θ) = 1$

$∴$ $f (θ) \in [1, 3] \Rightarrow α = 1, β = 3$

Sum of infinite G.P. with first term 64 and common ratio

$\frac{1}{3} = \frac{64}{1 - \frac{1}{3}} = 32 \times 3 = 96$

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