Question 9 - Jee advanced Math 2022 P1 Questions with Solutions

 Consider the equation

\(\int_{1}^{e} \frac{(\log_{e}{x})^{\frac{1}{2}}}{x(a - (\log_{e}{x})^\frac{3}{2})^{2}}dx = 1, \: \: a \in (-\infty, 0) U (1, \infty)\).

Which of the following statements is/are true?

(A) No \(a\) satisfies the above equation

(B) An integer \(a\) satisfies the above equation

(C) An irrational number \(a\) satisfies the above equation

(D) More than one \(a\) satisfy the above equation


Sol : 

\(\int_{1}^{e} \frac{(\log_{e}{x})^{\frac{1}{2}}}{x(a - (\log_{e}{x})^\frac{3}{2})^{2}}dx = 1\)

\(\implies \int_{1}^{e} \frac{(\log_{e}{x})^{\frac{1}{2}}\times \frac{1}{x}}{(a - (\log_{e}{x})^\frac{3}{2})^{2}}dx = 1\)

Let \((\log_{e}{x})^\frac{3}{2} = t\)

\(\implies (\frac{3}{2}(\log_{e}{x})^\frac{1}{2} \times \frac{1}{x})dx = dt\)

\(\implies ((\log_{e}{x})^\frac{1}{2} \times \frac{1}{x})dx = \frac{2}{3}dt\)

\(x = 1 \implies (\log_{e}{1})^\frac{3}{2} = 0 = t\)

\(x = e \implies (\log_{e}{e})^\frac{3}{2} = 1 = t\)

\(\implies \int_{0}^{1} \frac{\frac{2}{3}dt}{(a - t)^{2}}= 1\)

Let \(u = a - t\)

\(\implies du = - dt\)

\(t = 0 \implies u = a\)

\(t = 1 \implies u = a - 1\)

\(\implies -\frac{2}{3} \int_{a}^{a-1} u^{-2} du = 1\)

\(\implies -\frac{2}{3} [\frac{u^{-2+1}}{-2+1}]_{a}^{a-1} = 1\)

\(\implies \frac{2}{3} [\frac{1}{u}]_{a}^{a-1} = 1\)

\(\implies \frac{2}{3}[\frac{1}{a - 1} - \frac{1}{a}] = 1\)

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

\(\implies 3a^{2} - 3a - 2 = 0\)

Therefore,

\(a = \frac{-(-3) \pm \sqrt{9 + 24}}{6}\)

\(\implies a = \frac{3 \pm \sqrt{33}}{6}\)

\(\sqrt{33} > \sqrt{25} = 5\)  and \(-\sqrt{33} < -\sqrt{25} = -5\)

\(\implies 3 + \sqrt{33} > 3 + 5 = 8\) and \(3 - \sqrt{33} < 3 - 5 = -2\)

\(\implies \frac{3 + \sqrt{33}}{6} > \frac{8}{6}\) and \(\frac{3 - \sqrt{33}}{6} < \frac{-2}{6}\)

\(\implies \frac{3 + \sqrt{33}}{6} > 1\) and \(\frac{3 - \sqrt{33}}{6} < 0\)

Both the values are in the interval \(a \in (-\infty, 0) U (1, \infty)\).

Also, both \(a = \frac{3 \pm \sqrt{33}}{6}\) are irrational numbers.

Therefore, the correct options are C and D.

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