If the range of f(θ)=(sin ⁴ θ+3 cos ² θ)/(sin ⁴ θ+ cos ² θ), θ R is [α, β], then the sum of the infinite G.P., whose first term is 64 and the common ratio is (α)/(β), is equal to _ _ _ _ _.

Sets, Relations and Functions: JEE Main 2024 Mathematics question

JEE Main 2024 · 8 Apr, Morning shift · Q31

JEE Main 20248 Apr, Morning shiftMathematicsMediumSets, Relations and Functions168× since 2002

If the range of

f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}

[\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 __________.

Your answerType your answer. Integers and decimals are accepted.