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8. Sequences and Series
hard
If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite $G.P.$, whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to...........
A
$96$
B
$46$
C
$27$
D
$52$
(JEE MAIN-2024)
Solution
$ f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta} $
$ f(\theta)=1+\frac{2 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta} $
$ f(\theta)=\frac{2 \cos ^2 \theta}{\cos ^4 \theta-\cos ^2 \theta+1}+1 $
$ f(\theta)=\frac{2}{\cos ^2 \theta+\sec ^2 \theta-1}+1 $
$ \left.f(\theta)\right|_{\min .}=1 $
$ f(\theta)_{\max }=3 $
$ S=\frac{64}{1-1 / 3}=96$
Standard 11
Mathematics
