Question

The equation $x^2-2=\left[\sin x\right],$ where $[.]$ denotes the greatest integer function, has

• A. infinity many roots
• B. exactly one integer root
• C. exactly one irrational root
• D. exactly two roots

Answer 1

Answer: (B), (C), (D)

The correct options are
B exactly one integer root
C exactly one irrational root
D exactly two roots

Given equation $x^2-2=\left[\sin x\right]$
Now, when $x\in [0,\frac{\pi }{2}),\left[\sin x\right]=0$
At $x=\frac{\pi }{2},\left[\sin x\right]=1$
$x\in \left(\frac{\pi }{2},\pi \right],\left[\sin x\right]=0$
$x\in [-\frac{\pi }{2},0),\left[\sin x\right]=-1$

Case I: Taking $RHS=0$
So, LHS i.e. parabola is $0$ at $x=\sqrt{2}$
So, the point of intersection will be $x=\sqrt{2}$
Case II: Taking $RHS=1$
So, LHS is $1$ at $x=\sqrt{3}$
So, no point of intersection in this case.
Case III: $RHS=-1$
So, LHS should be $-1$ which is possible at $x=-1$
Hence , one point of intersection
So, exactly two roots , one integer and one irrational.