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
$\text { Let } a-\left(\log _e x\right)^{3 / 2}=t$
$\frac{\left(\log _e x\right)^{1 / 2}}{x} d x=-\frac{2}{3} d t$
$=\frac{2}{3} \int_a^{a-1} \frac{-d t}{t^2}=\frac{2}{3}\left(\frac{1}{t}\right)_a^{a-1}=1$
$\frac{2}{3 a(a-1)}=1$
$3 a^2-3 a-2=0$
$a=\frac{3 \pm \sqrt{33}}{6}$
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$f_1(x)=\left\{\begin{array}{lll}|x| & \text { if } & x<0, \\ e^x & \text { if } & x \geq 0 ;\end{array}\right.$
$f_2(x)=x^2$
$f_3(x)=\left\{\begin{array}{ccc}\sin x & \text { if } & x < 0, \\ x & \text { if } & x \geq 0\end{array}\right.$ and
$f_4(x)=\left\{\begin{array}{ccc}f_2\left(f_1(x)\right) & \text { if } & x < 0, \\ f_2\left(f_1(x)\right)-1 & \text { if } & x \geq 0\end{array}\right.$
| List $I$ | List $II$ |
| $P.$ $ f_4$ is | $1.$ onto but not one-one |
| $Q.$ $f_3$ is | $2.$ neither continuous nor one-one |
| $R.$ $f _2 \circ f _1$ is | $3.$ differentiable but not one-one |
| $S.$ $ f_2$ is | $4.$ continuous and one-one |
Codes: $ \quad P \quad Q \quad R \quad S $