Question

Let $\alpha$ and $\beta$ be any two positive values of $x$ for which $2\cos x,\left|\cos x\right|,$ and $1-3{\cos }^{2}x$ are in G.P. The minimum value of $|\alpha -\beta |$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. None of these

Answer 1

Answer: (D)

None of these

According to the given condition, ${\cos }^{2}x=2\cos x(1-3{\cos }^{2}x)$
$\Rightarrow \cos x(6{\cos }^{2}x+|\cos x|-2)=0$
$\Rightarrow \cos x=0$ or $6{\cos }^{2}x+4|\cos x|-3|\cos x|-2=0$
$\Rightarrow \cos x=0$ or $|\cos x|=\frac{1}{2}$
So, $\alpha =\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2}...$
$\beta =\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3}$
Hence the minimum difference is $\frac{\pi }{6}$