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STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
Consider the equation $\int_1^e \frac{\left(\log _e x \right)^{1 / 2}}{ x \left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty)$.

Which of the following statements is/are $TRUE$ ?

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

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

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

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

  • ✓
    $C,D$
  • B
    $C,B$
  • C
    $C,A$
  • D
    $A,B,C$
✓

Answer

Correct option: A.
$C,D$
a
$\int_1^e \frac{\left(\log _e x\right)^{1 / 2}}{x\left(a-\left(\log _e x\right)^{3 / 2}\right)^2}=1$

$\text { Let } a-\left(\log _e x\right)^{3 / 2}=t$

$\frac{\left(\log _e x\right)^{1 / 2}}{x} d x=-\frac{2}{3} d t$

$=\frac{2}{3} \int_a^{a-1} \frac{-d t}{t^2}=\frac{2}{3}\left(\frac{1}{t}\right)_a^{a-1}=1$

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

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

$a=\frac{3 \pm \sqrt{33}}{6}$

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