Question

If $x \in(-\pi, \pi)$ such that $y=1+|\cos x|+\left|\cos ^{2} x\right|+$ $\left|\cos ^{3} x\right|+\ldots .$ and $8^{y}=64$, then the number of values of x satisfying the equations is :

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

Here series is in G.P. $a =1, r =|\cos x |$
$\therefore y=\frac{1}{1-|\cos x|}$
$\Rightarrow 8^{\frac{1}{1-|\cos x|}=64=8^{2}} $
$\Rightarrow 1-|\cos x|=\frac{1}{2} $
$\Rightarrow |\cos x|=\frac{1}{2} $
$\Rightarrow \cos x=\pm \frac{1}{2}$
$\Rightarrow x=\pm \frac{\pi}{6}, \pm \frac{5 \pi}{6} $