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JEE Main 20248 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Functions

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Functions

If the range of f(θ)=( ^4 θ+3 ^2 θ)/( ^4 θ+ ^2 θ), θ ∈ ℝ is [α, β]...

If the range of f(θ)=sin⁡4θ+3cos⁡2θsin⁡4θ+cos⁡2θ,θ∈Rf(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R} is [α,β][\alpha, \beta], then the sum of the infinite G.P., whose first term is 64 and the common ratio is αβ\frac{\alpha}{\beta}, is equal to________

Correct answer

0

Step-by-step solution

f(θ)=sin⁡4θ+3cos⁡2θsin⁡4θ+cos⁡2θf(θ)=1+2cos⁡2θsin⁡4θ+cos⁡2θf(θ)=2cos⁡2θcos⁡4θ−cos⁡2θ+1+1f(θ)=2cos⁡2θ+sec⁡2θ−1+1f(θ)∣min. =1f(θ)max. =3S=641−1/3=96\begin{aligned} & 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|_{\text{min. }}=1 \\ & f(\theta)_{\text{max. }}=3 \\ & S=\frac{64}{1-1 / 3}=96\end{aligned}
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