Question

The most general solutions of $2^{1+\left|\cos x\right|+{\cos }^2+{\left|\cos x\right|}^3+...to\infty }=4$ are given by

• A. $n\pi \pm \frac{\pi }{3},nϵZ$
• B. $2n\pi \pm \frac{\pi }{3},nϵZ$
• C. $2n\pi \pm \frac{2\pi }{3},nϵZ$
• D. none of these

Answer 1

Answer: (A)

$n\pi \pm \frac{\pi }{3},nϵZ$

Given, $2^{1+\left|\cos x\right|+{\cos }^2+{\left|\cos x\right|}^3+...to\infty }=4$
$\Rightarrow 1+\left|\cos x\right|+{\cos }^2+{\left|\cos x\right|}^3+...to\infty =2$
$\Rightarrow \frac{1}{1-|\cos x|}=2$
$\Rightarrow |\cos x|=\frac{1}{2}$
$\Rightarrow \cos x=\frac{1}{2}$ or $\cos x=-\frac{1}{2}$
$\Rightarrow \cos x=\cos \frac{\pi }{3},\cos x=\cos \frac{2\pi }{3}$
The general solutions are $x=2n\pi \pm \frac{\pi }{3},x=2n\pi \pm \frac{2\pi }{3}$
This can also be written as $x=n\pi \pm \frac{\pi }{3}$