A farmer has a land in the shape of a triangle with vertices at P(0, 0), Q (1, 1) and R(2, 0). 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 30% of the area of ΔPQR, then the value of n is

4

It is evident from Fig. that

$\int\limits_0^1(y_2-y_1)dx$ = 30% of ΔPQR

$⇒\int\limits_0^1(x-x^n)dx=\frac{30}{100}×\frac{1}{2}×2×1$

$⇒\left[\frac{x^2}{2}-\frac{x^{n+1}}{n+1}\right]_0^1=\frac{3}{10}$

$⇒\frac{1}{2}-\frac{1}{n+1}=\frac{3}{10}⇒\frac{1}{n+1}=\frac{1}{2}-\frac{3}{10}=\frac{1}{5}⇒n+1=5⇒n=4$