Question

If the function $f\left(x\right)=\left[\frac{(x-3{)}^2}{a}\right]\sin (x-3)+a\cos (x-3)$ is continuous in $[4,8]$, then the range of $a$ is
($[.]$ denotes the greatest integer function)

• A. $a<25$
• B. $a>25$
• C. $a<30$
• D. $a>30$

Answer 1

The correct option is B $a>25$

We know that $\sin (x-3)$ and $\cos (x-3)$ are continuous for all values of $x$.
$\left[x^2\right]$ is discontinuous at integral points.
$f\left(x\right)$ is continuous in $[4,8]$ if $\left[\frac{(x-3{)}^2}{a}\right]=0\forall x\in [4,8]$
Now, $(x-3{)}^2\in [1,25]$ for $x\in [4,8]$
$\Rightarrow a>25$ for $\left[\frac{(x-3{)}^2}{a}\right]=0$