Question

Let $f:R\to R$ be a function defined by $f\left(x\right)=\left\{\left|\cos x\right|\right\},$ where $\left\{x\right\}$ represents fractional part of $x$. Let $S$ be the set containing all real values $x$ lying in the interval $[0,2\pi ]$ for which $f\left(x\right)\ne \left|\cos x\right|.$ Then, number of elements in the set $S$ is

• A. $0$
• B. $1$
• C. $3$
• D. infinite

Answer 1

The correct option is C $3$

$f\left(x\right)\ne \left|\cos x\right|$ is true only when

Function inside the fractional part must be an integer.

$\cos x$ has a maximum integer value $=1$

$\left|\cos x\right|=1$

As the interval is $[0,2\pi ]$

$\Rightarrow x=0,\pi ,2\pi$

So, the number of elements $=3$