Question

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 ____.

Answer 1

Correct Answer: 96

The given function is:
\[ f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}. \]
Substitute \(\cos^2\theta = x\), with \(x \in [0, 1]\):
\[ f(x) = \frac{\sin^4\theta + 3x}{\sin^4\theta + x}. \]
Simplify:
\[ f(x) = \frac{2x}{x + 1} + 1. \]
The range of \(f(x)\) can be computed as:
\[ f_{\min} = 1, \quad f_{\max} = 3. \]
Thus:
\[ \alpha = 1, \quad \beta = 3. \]
The infinite geometric series is:
\[ S = \frac{\text{first term}}{1 - \text{common ratio}}. \]
Substitute:
\[ S = \frac{64}{1 - \frac{1}{3}} = \frac{64}{\frac{2}{3}} = 64 \cdot \frac{3}{2} = 96. \]
Final Answer: 96.