Question

Let $g:[-2,2]\to R$ where $g\left(x\right)=x^3+\tan x+\left[\frac{x^2+1}{P}\right]$ be an odd function , then the value of the parameter $P$ satisfies
(Note : $\left[a\right]$ denotes the greatest integer less than or equal to $a$)

• A. $-5<P<5$
• B. $P<5$
• C. $-5<P<0$
• D. $P>5$

Answer 1

The correct option is D $P>5$

Let $h\left(x\right)=\left[\frac{x^2+1}{P}\right]$.

Since, $x^3$ and $\tan x$ are odd functions $g\left(x\right)$ is odd

if and only if $h\left(x\right)$ is an odd function or $h\left(x\right)=0$ for all $x$.

Now, $-2\le x\le 2$ implies that $1\le x^2+1\le 5$.

Hence, for $P\le 5$, the function $h\left(x\right)$ will assume some nonzero integral values dependant on $x$. But then $h\left(x\right)$ can not be an odd function.

For $P>5$ the function $h\left(x\right)=0$ for all $x$ and hence $g\left(x\right)$ is an odd function.