Sets, Relations & Functions
Range of Trigonometric Expression — Infinite GP
nta_pyq_2024_apr
Grade 11
Question:
If the range of $f(\theta)=\dfrac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}$, $\theta\in\mathbb{R}$, is $[\alpha,\beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\dfrac{\alpha}{\beta}$, is equal to ________.
Step-by-Step Solution
Key Concept: Simplify $f(\theta)$: let $t=\cos^2\theta\in[0,1]$. $f=1+\frac{2t}{(1-t)^2+t}$. Min at $t=0$: $f=1$, max at $t=1$: $f=3$. $[\alpha,\beta]=[1,3]$.
$\alpha=1$, $\beta=3$, ratio $=1/3$. Sum $=64\div(2/3)=96$.
Correct Answer: 96