JEE Advanced 2014 · Paper 1, Mathematics question 58

The graph of $f$. $\cos^{-1}(\cos x)$ is a triangle wave:
$$f(x)=\begin{cases}x&0\le x\le\pi\\2\pi-x&\pi\le x\le2\pi\\x-2\pi&2\pi\le x\le3\pi\\4\pi-x&3\pi\le x\le4\pi\end{cases}$$

Line. $y=1-\frac x{10}$ falls from 1 to $1-0.4\pi\approx-0.26$.

Solve on each piece.

• $x=1-\frac x{10}$: $x=\frac{10}{11}\approx0.91$, in $[0,\pi]$. ✓
• $2\pi-x=1-\frac x{10}$: $x=\frac{2\pi-1}{0.9}\approx5.87$, in $[\pi,2\pi]$. ✓
• $x-2\pi=1-\frac x{10}$: $x=\frac{2\pi+1}{1.1}\approx6.62$, in $[2\pi,3\pi]$. ✓
• $4\pi-x=1-\frac x{10}$: $x=\frac{4\pi-1}{0.9}\approx12.85>4\pi$. ✗

Answer: 3