JEE Advanced 2022 Paper 1 Q09 Mathematics Integration & Differential Equations Definite Integrals Medium

JEE Advanced 2022 Paper 1 · Q09 · Definite Integrals

Consider the equation $\displaystyle\int_{1}^{e}\dfrac{(\log_{e}x)^{1/2}}{x\,\big(a-(\log_{e}x)^{3/2}\big)^{2}}\,dx=1$, $a\in(-\infty,0)\cup(1,\infty)$. Which of the following statements is/are TRUE?

  1. A. No $a$ satisfies the above equation
  2. B. An integer $a$ satisfies the above equation
  3. C. An irrational number $a$ satisfies the above equation
  4. D. More than one $a$ satisfy the above equation
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:C, D

Solution

Let $t=(\log_{e}x)^{3/2}$, then $dt=\dfrac{3}{2}\dfrac{(\log_{e}x)^{1/2}}{x}dx$, so $\dfrac{(\log_{e}x)^{1/2}}{x}dx=\dfrac{2}{3}dt$. When $x=1, t=0$; when $x=e, t=1$. Integral becomes $\displaystyle\int_{0}^{1}\dfrac{2/3}{(a-t)^{2}}dt=\dfrac{2}{3}\!\left[\dfrac{1}{a-t}\right]_{0}^{1}=\dfrac{2}{3}\!\left(\dfrac{1}{a-1}-\dfrac{1}{a}\right)=\dfrac{2}{3}\cdot\dfrac{1}{a(a-1)}=1$. So $a(a-1)=2/3\Rightarrow 3a^{2}-3a-2=0\Rightarrow a=\dfrac{3\pm\sqrt{9+24}}{6}=\dfrac{3\pm\sqrt{33}}{6}$. Both roots satisfy $a\in(-\infty,0)\cup(1,\infty)$ and are irrational. Hence (C) and (D) are TRUE.

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