Question

If $f:[-15,15]\to R$, defined by $f\left(x\right)=\left[\frac{x^2}{a}\right]\sin x+\cos x$ ( where $[\cdot ]$ denotes greatest integer function) is an even function then set of values of $a$ belongs in interval

• A. $(225,\infty )$
• B. $R$
• C. $(0,225)$
• D. $\varphi$

Answer 1

The correct option is A $(225,\infty )$

Note that, if $\left[\frac{x^2}{a}\right]\ge 1$ then $f\left(x\right)=k\sin \left(x\right)+\cos \left(x\right)$, where $k$ is an positive integer greater than $1$.

Then in this case, the function will not be an even function for any value of $a$.

Hence the function will be an even function if and only if $\left[\frac{x^2}{a}\right]=0$

Hence, $x^2<a$

Since $x\in [-15,15]$, we have $a>225$.