JEE Advanced 2022 · Paper 1, Mathematics question 9

Substitution. Put $t=(\log_ex)^{3/2}$, so $dt=\frac32(\log_ex)^{1/2}\frac{dx}{x}$:
$$\int_0^1\frac{2}{3}\frac{dt}{(a-t)^2}=\frac23\left[\frac{1}{a-t}\right]_0^1=\frac23\cdot\frac{1}{a(a-1)}.$$

Solve. Setting this equal to 1:
$$a^2-a-\frac23=0\implies a=\frac{1\pm\sqrt{11/3}}2\approx1.457,\ -0.457.$$

Check. The first root lies in $(1,\infty)$ and the second in $(-\infty,0)$, so both are allowed. Both are irrational, and there are two of them.

(C) and (D) are true; (A) and (B) are false.