Question

If the line x = α divides the area of region R = {(x, y) ∈ R² : x³ ≤ y ≤ x, 0 ≤ x ≤ 1}  into two equal parts, then

(A) 0 <  α ≤ 1/2

(B) 1/2< α < 1

(C) 2α⁴ - 4α² + 1 = 0

(D) α⁴ + 4α² - 1 = 0 

Answer 1

Answer is (B), (C)

Let us consider y = x³ and y = x. Then the area between these two curves in region 0 ≤ x ≤ 1  is

It is given that the line x = α divides the area under the curve into two equal parts. Therefore,

Now, let us consider the equation 2α⁴ – 4α² + 1 = 0.

Let α² = u. Therefore,