Question

If $g\left(x\right)=\left[\sin x\right]$ and $f\left(x\right)=1-x^2,$ then which of the following is/are CORRECT ?
(where $[.]$ denotes greatest integer function)

• A. $f\left(x\right)=g\left(x\right)$ has $1$ solution.
• B. $f\left(x\right)=g\left(x\right)$ has $2$ solutions.
• C. The sum of the solution(s) of $f\left(x\right)=g\left(x\right)$ is positive.
• D. The sum of the solution(s) of $f\left(x\right)=g\left(x\right)$ is negative.

Answer 1

Answer: (B), (D)

The sum of the solution(s) of $f\left(x\right)=g\left(x\right)$ is negative.

Drawing the graph of $g\left(x\right)=\left[\sin x\right]$ and $f\left(x\right)=1-x^2$


The number of solutions of the given equation is $2.$
Solution occurs when $\left[\sin x\right]=0,-1$, then
$1-x^2=0\Rightarrow x=\pm 11-x^2=-1\Rightarrow x=\pm \sqrt{2}$
From the graph, the solutions are $x=-\sqrt{2},1$
Sum of solutions $=1-\sqrt{2}<0$