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  4. If n be the number of distinct solutions of the equation cos- 1|x|+cos- 1|2 x|=π , then the value of (1/n) is equal to

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Q. If $n$ be the number of distinct solutions of the equation $cos^{- 1}\left|x\right|+cos^{- 1}\left|2 x\right|=\pi ,$ then the value of $\frac{1}{n}$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

$x \geq 0$
$\cos ^{-1} x+\cos ^{-1} 2 x=\pi .$
$2 x=\cos \left(\pi-\cos ^{-1} x\right)$
$=-\cos \left(\cos ^{-1} x\right)$
$=-x \quad \Rightarrow \quad x=0$
$x \leq 0$
$\cos ^{-1}(-x)+\cos ^{-1}(-2 x)=\pi$
$\pi-\cos ^{-1} x+\pi-\cos ^{-1}(2 x)=\pi$
$\cos ^{-1} x+\cos ^{-1}(2 x)=\pi$