Question
For all permissible values of $x$, consider $y = \frac{sin3x(cos6x + cos4x)}{sinx(cos8x + cos2x)}$ and range of $y$ is $( - \infty,a) \cup (b,\infty)$. If $2b$ is the first term of G.P. and ' $a$ ' is its common ratio, then ( $s_{\infty}$ denotes the sum of infinite terms of G.P.)
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Solution
Sol. [Key Concept: Range of the trig expression gives the G.P. first term/ratio; then sum to infinity.]
$$y = \frac{tan3x}{tanx} = \frac{3 - \tan^{2}x}{1 - 3\tan^{2}x} = \frac{8}{3\left( 1 - 3\tan^{2}x \right)} + \frac{1}{3}$$
$$\begin{matrix} \Rightarrow \ & y \in \left( - \infty,\frac{1}{3} \right) \cup (3,\infty) \\ & a = \frac{1}{3},b = 3,s_{\infty} = \frac{6}{1 - \frac{1}{3}} = 9 \end{matrix}$$
✘ (a) $b - a = \frac{83}{\neq}\frac{10}{3}$
✔ (b) $3a + b = 1 + 3 = 4$
✔ (c) $S_{\infty} = \frac{2b}{1 - a} = \frac{6}{2/3} = 9$
✔ (d) $\frac{27}{10}(a + b) = \frac{27}{10} \cdot \frac{10}{3} = 9$
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