Sequences & Series
Sequence and Series
Allen Star Batch
Grade 11
Question:
For all permissible value of $x$, consider $y = \frac{\sin 3x(\cos 6x + \cos 4x)}{\sin x(\cos 8x + \cos 2x)}$ and range of $y$ is $(−\infty, a) \cup (b, \infty)$. If 2b is the first terms of G.P. and 'a' is its common ratio, then: ($S_{\infty}$ denotes the sum of infinite terms of G.P.)
$b - a = -\frac{10}{3}$
$3a + b = 4$
$S_{\infty} = 9$
$S_{\infty} = \frac{27}{10}(a+b)$
Step-by-Step Solution
Key Concept: Combine multiple trigonometric fractions and analyze their range by finding critical points where the derivative equals zero.
Starting with the given expression $y = \frac{\tan 3x}{\tan x} - \frac{3 - \tan^2 x}{1 - 3\tan^2 x} = \frac{8}{3(1 - 3\tan^2 x)} + \frac{1}{3}$, we use trigonometric identities to simplify. The range is found by analyzing the behavior of the combined expression, yielding $y \in (-\infty, -\frac{1}{3}] \cup [3, \infty)$. The critical parameters are $a = \frac{1}{3}$, $b = 3x_c$, and $x_c = 9$.
Correct Answer: 2,3,4