Consider the equation ∫1elogex^12xa-logex^322dx=1,a∈-∞,0∪1,∞. Which of the following statements is/are TRUE?

Correct answer: (C) An irrational number a satisfies the above equation; (D) More than one a satisfy the above equation. Let I=∫1e(lnx^12dx)/(xa-lnx^322) Now put a-lnx^32=t ⇒-3/2lnx^12·1/xdx=dt So, the integral becomes I=∫aa-1(-23dtt^2⇒I=-23t^(-2+1)-2+1aa-1⇒I=23taa-1⇒I=231a-1-1aNow given I=231aa-1=1⇒2=3a^2-3a⇒3a^2-3a-2=0⇒ a=3±√(9-43-2))/6 So, a=(3+√33)/6,(3-√33)/6

Q27 / 54 · JEE Advanced 2022 (Paper 1) · Mathematics

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

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

Solution

Let $I = \int_{1}^{e} \frac{{(\ln x)}^{\frac{1}{2}} d x}{x {(a - {(\ln x)}^{\frac{3}{2}})}^{2}}$

Now put $a - {(\ln x)}^{\frac{3}{2}} = t$

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

So, the integral becomes $I = \int_{a}^{a - 1} \frac{(- \frac{2}{3}) d t}{t^{2}}$

$\Rightarrow I = (- \frac{2}{3}) {\frac{t^{- 2 + 1}}{- 2 + 1}}_{a}^{a - 1}$

$\Rightarrow I = {\frac{2}{3 t}}_{a}^{a - 1}$

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

Now given $I = (\frac{2}{3}) \frac{1}{a (a - 1)} = 1$

$\Rightarrow 2 = 3 a^{2} - 3 a$

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

$\Rightarrow a = \frac{3 \pm \sqrt{9 - 4 (3) (- 2)}}{6}$

So, $a = \frac{3 + \sqrt{33}}{6}, \frac{3 - \sqrt{33}}{6}$

Hence, the correct answer is options: 'C', 'D'.

Consider the equation∫1elogex12xa-logex322dx=1,a∈-∞,0∪1,∞.Which of the following statements is/are TRUE?

Solution

More Definite Integration Questions

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