Question

The equation \(x^{\left(\log _{3} x\right)^{2}-\frac{9}{2} \log _{3} x+5}=3 \sqrt{3}\) has
(A) at least one real root
(B) exactly one real root
(C) exactly one irrational root
(D) complex roots

Answer 1

Ans: \((A, C)\)
Hint \(: 2 t^{3}-9 t^{2}+10 t-3=0, t=\log _{3} x\)
\(\Rightarrow(t-3)(t-1)\left(t-\frac{1}{2}\right)=0\)
\(\Rightarrow x=3^{3}, 3^{1}, 3^{1 / 2}\)