Question

If $f:[-20,20]\to R$, defined by $f\left(x\right)=\left[\frac{x^2}{a}\right]\sin x+\cos x,$
(where $[.]$ denotes the greatest integer function) is an even function then set of values of ${}^{\prime }{a}^{\prime }$ is given by

• A. $nullset$
• B. $R$
• C. $[0,400]$
• D. $(400,\infty )$

Answer 1

The correct option is D $(400,\infty )$

$\text{As}f\left(x\right)\text{is even function,}$

$\Rightarrow f\left(x\right)=f(-x)$

$\Rightarrow \left[\frac{x^2}{a}\right]\sin$

$x=-\left[\frac{x^2}{a}\right]\sin x+\cos x$

$\Rightarrow \left[\frac{x^2}{a}\right]=0$

$\Rightarrow \frac{x^2}{a}\in [0,1)$

$x\in [-20,20]\Rightarrow x^2$

$\in [0,400]$

$x^2<1\Rightarrow a>x^2(\because x^2<400)$

$a\Rightarrow a>400\Rightarrow a\in (400,\infty )$

option $D$ is correct.