Verified Solution • 2025–26 Session moderate

QJEEAdv17Mae8d4: JEE Advanced 2017 Mathematics - Class 12 JEEAdv17Mae8d4 Solution: If the line x = divides the area of region R = {(x, y) R 2 : x…

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

Multiple Choice Options

  • A 2 α\alpha 4 −- 4 α\alpha 2 + 1 =0
  • B α\alpha 4 + 4 α\alpha 2 −- 1 =0
  • C 12<α<1{1 \over 2} < \alpha < 1
  • D 0 < α\alpha ≤\le 12{1 \over 2}

Step-by-Step Verified Solution

∫01(x−x3)dx=2∫0αx−x3)dx\int_0^1 {(x - {x^3})dx = 2\int_0^\alpha {x - {x^3})dx} }

14=2(α22−α24){1 \over 4} = 2\left( {{{{\alpha ^2}} \over 2} - {{{\alpha ^2}} \over 4}} \right)

2α2−4α2+1=02{\alpha ^2} - 4{\alpha ^2} + 1 = 0

α2=4−16−84{\alpha ^2} = {{4 - \sqrt {16 - 8} } \over 4} ( ∵\because α\alpha ∈\in (0, 1))

∵\because α2=1−12{\alpha ^2} = 1 - {1 \over {\sqrt 2 }}

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