Question

The number of real solutions of the equation $1-x=\left[\cos x\right]$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

For $x=0,\left[cosx\right]=1$
Expression becomes $1-x=1\Rightarrow x=0$
For $x\in (0,\frac{\pi }{2}),\left[cosx\right]=0$
Expression becomes $1-x=0\Rightarrow x=1$, which is in range.
For $x\in (\frac{\pi }{2},\pi ),\left[cosx\right]=-1$
Expression becomes $1-x=-1\Rightarrow x=2$, which is also in range.
For $x\in (\pi ,\frac{3\pi }{2}),\left[cosx\right]=0$
Expression becomes $1-x=0\Rightarrow x=1$, which is not in the considered range.
For later values of $x$, it will not satisfy the expression.
Therefore only $3$ values of $x$.