Question

Let $f:[-25,25]\to R$ where $f\left(x\right)$ is defined by $f\left(x\right)={\sin }^{n}x+\left[\frac{x^4}{a}\right]$ be an odd function, if $n$ is odd. Then the set of values of the parameter $a$ is/are $\left(\right[\cdot ]$ denotes greatest integer function$)$

• A. $(-25,25)-\left\{0\right\}$
• B. $[390625,\infty )$
• C. $(-25,0)$
• D. $(390625,\infty )$

Answer 1

Answer: (D)

Since $n$ is odd, therefore ${\sin }^n(-x)=-{\sin }^{n}x$
For an odd function, $f\left(x\right)=-f(-x)$
$-({\sin }^n(-x)+\left[\frac{{(-x)}^4}{a}\right])={\sin }^{n}x+[\frac{x^4}{a}]$
$\Rightarrow {\sin }^n\left(x\right)-\left[\frac{x^4}{a}\right]={\sin }^{n}x+\left[\frac{x^4}{a}\right]$
$\Rightarrow \left[\frac{x^4}{a}\right]=0$
$\Rightarrow \left[\frac{x^4}{a}\right]<1$
$\Rightarrow a>{25}^4$ since $x\in [-25,25]$
$\Rightarrow a>390625$