Question

Let $f (x): cos \,x ( cos\,x + sin\,x) = [x]$, where $[\cdot]$ represent the greatest integer function, the interval for which $x$ is domain of $f (x)$, is

• A. $[0, 1) \cup [1, 2]$
• B. $ 0 le x < 1 \cup 1 \le x < 2$
• C. $0 < x < 1 \cup 1 < x < 2$
• D. $- 1 < x < 1 \cup 1 \le x < 2$

Answer 1

Answer: (B)

$f(x): \cos x(\cos x+\sin x)=[x]$
$\Rightarrow \cos ^{2} x+\sin x \cos x=[x]$
$\Rightarrow \cos 2 x+\sin 2 x=2[x]-1$ (Multiply by 2 on both sides
and use formula $\left.\cos 2 x=2 \cos ^{2} x-1\right)$
$\Rightarrow \sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)=2[x]-1$
$\Rightarrow -\sqrt{2} \leq 2[x]-1 \leq \sqrt{2}$
$ \Rightarrow \frac{1-\sqrt{2}}{2} \leq[x] \leq \frac{\sqrt{2}+1}{2}$
$\Rightarrow [x]=0,1 \therefore [x]=0,[x]=1$
$\Rightarrow x \in[0,1)$ and $x \in[1,2)$