Q108 / 155 · WBJEE 2020 · Mathematics

The equation $x^{\left(\log _{3} x\right)^{2}-\frac{9}{2} \log _{3} x+5}=3 \sqrt{3}$ has

  • Correct answer
    A
    at least one real root
  • B
    exactly one real root
  • Correct answer
    C
    exactly one irrational root
  • D
    complex roots

Solution

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

Hence, the correct answer is options: 'A', 'C'.

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