Question

The equation $x^2$ – 2 = [sin x], where [·] denotes the greatest integer function, has

• A. infinitely many roots
• B. only one root which is an integer
• C. only one root which is irrational
• D. exactly two roots

Answer 1

The correct option is D

exactly two roots

We have [sin x] = –1, 0, 1
So, we have the following cases
Case – I :- when [sin x] = –1 In this case, we have $x^2$ – 2 = –1 $\Rightarrow x\pm 1$
$\therefore$ x = –1 is the solution in this case
Case – II :- when [sin x] = 0 In this case, we have $x^2-2=0\Rightarrow x\pm \sqrt{2}$
But, $\left[sin\sqrt{2}\right]=0\left[sin\right(-\sqrt{2}\left)\right]=-1,\therefore x=\sqrt{2}$ is the solution in this case
Case – III :- when [sin x] = 1 In this case, we have $x^2-2=1\Rightarrow x\pm \sqrt{3}$
But $\left[sin\sqrt{3}\right]=0\left[sin\right(-\sqrt{3}\left)\right]=-1$ Therefore, there is no solution in this case.
Hence, the given equation has two solutions only, namely, x = –1 and x = $\sqrt{2}$