Question

If $g:[2:2]\to R$ where $g\left(x\right)=x^3+tanx+\left[\frac{x^2+1}{p}\right]$ is a odd function then the value of parametric P is where [.] denotes the Greatest integer function.

• A. $-5<p<5$
• B. $p<5$
• C. $p>5$
• D. None of these

Answer 1

The correct option is C $p>5$

$g\left(x\right)=x^3+tanx+\left[\frac{x^2+1}{p}\right]$

$g(-x)=(-x{)}^3+tan(-x)+[\frac{(-x{)}^2+1}{p}]$

$g(-x)=-x^3-tanx+\left[\frac{x^2+1}{p}\right]$

$g\left(x\right)+g(-x)=0$ because g(x) is a odd function

$\therefore x^3+tanx+\left[\frac{x^2+1}{p}\right]-x^3-tanx+\left[\frac{x^2+1}{p}\right]=0$

$\Rightarrow 2\left[\frac{(x^2+1)}{p}\right]=0$

$\Rightarrow 0\le \frac{x^2+1}{p}<1$ because $x\in [-2,2]$

$\Rightarrow 0\le \frac{5}{p}<1\Rightarrow p>5.$