A farmer $F_1$ has a land in the shape of a triangle with vertices at $P\left(\mathrm{0,\; 0}\right),Q\left(\mathrm{1,\; 1}\right)$ and $R\left(\mathrm{2,\; 0}\right).$ From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y=x^n, n>1$. If the area of the region taken away by the farmer $F_2$ is exactly $\mathrm{30\%}$ of the area of $∆PQR,$ then the value of $n$ is

Solution

Area $=\underset{0}{\overset{1}{\int }}\left(x-x^n\right)dx=\frac{3}{10}$
${\left[\frac{x^2}{2}-\frac{x^{n+1}}{n+1}\right]}_0^1=\frac{3}{10}$
$\frac{1}{2}-\frac{1}{n+1}=\frac{3}{10}$ $\therefore n+1=5$
$\Rightarrow n=4$

Hence, the correct answer is 4.