JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Area Under Curves

A farmer

F_{1}

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

P Q

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

△ P QR

, then the value of

n

is ____.

Answer: 4

Solution

Area of

△ P QR = \frac{1}{2} \times 2 \times 1 = 1

.

The line

P Q

has equation

y = x

for

x \in [0, 1]

.

Area between

y = x

and

y = x^{n}

\int_{0}^{1} (x - x^{n}) d x = [\frac{x ^{2}}{2} - \frac{x ^{n + 1}}{n + 1}]_{0}^{1} = \frac{1}{2} - \frac{1}{n + 1}

.

Setting this equal to

0.3 \times 1 = 0.3

\frac{1}{2} - \frac{1}{n + 1} = 0.3

, so

\frac{1}{n + 1} = 0.2

, giving

n + 1 = 5

, thus

n = 4

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Numerical answer