Question

If the area bounded by the curves $y=\left[k\right]x^2,y=\frac{\left[k\right]}{4}{x}^2$ and $2\le |x|\le 3$ is $19$, then $k$ lies in
(where [.] represent greatest integer function)

• A. $[2,3)$
• B. $[1,3)$
• C. $[0,3)$
• D. $[3,4)$

Answer 1

The correct option is A $[2,3)$


The required area
$19=2\underset{2}{\overset{3}{\int }}\left(\right[k]x^2-\frac{\left[k\right]}{4}{x}^2)dx\Rightarrow 19=2\underset{2}{\overset{3}{\int }}\frac{3\left[k\right]}{4}{x}^{2}dx\Rightarrow 19=\frac{\left[k\right]}{2}(27-8)\Rightarrow \left[k\right]=2\therefore k\in [2,3)$