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Previous Year Question (PYQs)

The number of real solutions of $\sqrt{1+\cos 2x}=\sqrt{2},\cos^{-1}(\cos x)$ in $\left[\frac{\pi}{2},\pi\right]$ is




Solution

$\sqrt{1+\cos 2x}=\sqrt{2},|\cos x|$. On $\left[\frac{\pi}{2},\pi\right]$, $\cos x\le 0$ so $|\cos x|=-\cos x$. LHS $=\sqrt{2},|\cos x|$, RHS $=\sqrt{2},|\cos x|$ only if $\cos^{-1}(\cos x)=|\cos x|$, which never holds for $x\in\left[\frac{\pi}{2},\pi\right]$. Hence no real solution.

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