Consider the equation int 1 e frac (log e x ) 1/2 x (a- (log e x ) 3/2 ) 2 d x=1, a in(-∞, 0) cup(1, ∞) . Which of the following statements is/are TRUE?

An irrational number a satisfies the above equation

More than one a satisfy the above equation

No a satisfies the above equation

An integer a satisfies the above equation

Mathematics · Definite Integration

JEE Advanced 2022 — Paper 1 — Question 9

Consider the equation $\int_{1}^{e} \frac{\left(\log _{e} x\right)^{\frac{1}{2}}}{x\left(a-\left(\log _{e} x\right)^{\frac{3}{2}}\right)^{2}} d x=1, a \in(-\infty, 0) \cup(1, \infty)$ . Which of the following

statements is/are TRUE?

Option A:
No $a$ satisfies the above equation
Option B:
An integer $a$ satisfies the above equation
Option C:
An irrational number $a$ satisfies the above equation
Correct
Option D:
More than one $a$ satisfy the above equation
Correct

Answer: C, D

Step-by-step solution

$\int_{1}^{e} \frac{(\ln x)^{1 / 2}}{x\left(a-(\ln x)^{3 / 2}\right)^{2}} d x=1$

$\mathrm{a}-(\ln \mathrm{x})^{3 / 2}=\mathrm{t}$

$-\frac{3}{2}(\ln x)^{1 / 2} \times \frac{1}{x} d x=d t$

$\int_{a}^{a-1} \frac{-\frac{2}{3} d t}{t^{2}}=1$

$\left[-\frac{2}{3} \times-\frac{1}{\mathrm{t}}\right]^{\mathrm{a}-1}=1$

$\frac{2}{3}\left[\frac{1}{a-1}-\frac{1}{a}\right]=1$

$\frac{a-a+1}{a(a-1)}=\frac{3}{2}$

$3\left(a^{2}-a\right)=2$

$3 a^{2}-3 a-2=0$

$\mathrm{a}=\frac{3 \pm \sqrt{9+24}}{6}=\frac{3 \pm \sqrt{33}}{6}$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2022
Paper: Paper 1
Subject: Mathematics
Chapter: Definite Integration
Topic: Methods of solving definite integrals(kings rule,odd even)