Question

Let $f:[-10,10]\to R,$ where $f\left(x\right)=\sin x+\left[\frac{x^2}{a}\right]$ be an odd function. Then the set of values of parameter $a$ is (where $[.]$ denotes greatest integer function)

• A. $[100,\infty )$
• B. $(-10,10)-\left\{0\right\}$
• C. $(0,10)$
• D. $(100,\infty )$

Answer 1

The correct option is D $(100,\infty )$

$\because f\left(x\right)$ is an odd function
$\therefore f\left(x\right)=-f(-x)$
$\Rightarrow \sin x+\left[\frac{x^2}{a}\right]=-(-\sin x+\left[\frac{x^2}{a}\right])$
$\Rightarrow \left[\frac{x^2}{a}\right]=0\forall x\in [-10,10]$
$\Rightarrow 0\le \frac{x^2}{a}<1\forall x\in [-10,10]$
$\Rightarrow a>100$