Question

If $x\in (-\pi ,\pi )$ such that $y=1+|\cos x|+\left|{\cos }^{2}x\right|+$ $\left|{\cos }^{3}x\right|+\dots .$ 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. $a=1,r=|\cos x|$
$\therefore y=\frac{1}{1-|\cos x|}$
$\Rightarrow 8^{\frac{1}{1-|\cos x|}=64=8^2}$
$\Rightarrow 1-|\cos x|=\frac{1}{2}$
$\Rightarrow |\cos x|=\frac{1}{2}$
$\Rightarrow \cos x=\pm \frac{1}{2}$
$\Rightarrow x=\pm \frac{\pi }{6},\pm \frac{5\pi }{6}$