Consider the equation ∫ 1 e log e x 1 2 x a - log e x 3 2 2 d x = 1 , a ∈ - ∞ , 0 ∪…

Consider the equation

∫1elogex12xa-logex322dx=1,a∈-∞,0∪1,∞.

Which of the following statements is/are TRUE?

  1. No a satisfies the above equation
  2. An integer a satisfies the above equation
  3. An irrational number a satisfies the above equation
  4. More than one a satisfy the above equation

Solution

Let I=∫1elnx12dxxa-lnx322

Now put a-lnx32=t

⇒-32lnx12·1xdx=dt

So, the integral becomes I=∫aa-1-23dtt2

⇒I=-23t-2+1-2+1aa-1

⇒I=23taa-1

⇒I=231a-1-1a

Now given I=231aa-1=1

⇒2=3a2-3a

⇒3a2-3a-2=0

⇒ a=3±9-43-26

So, a=3+336,3-336

Asked in: JEE Advanced 2022 (Paper 1)

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