JEE Main 2025 Jan 22 Shift 1, Mathematics Q9: Quadratic Equations

JEE Main2025Jan 22, Shift 1Mathematics

The product of all solutions of the equation $e^{5 (l o g_{e} x)^{2} + 3} = x^{8}$ , $x > 0$ , is :

Solution

Take $ln$ of both sides: $5 (ln x)^{2} + 3 = 8 ln x$ .

Let $t = ln x$ : $5 t^{2} - 8 t + 3 = 0$ .

By Vieta's, $t_{1} + t_{2} = \frac{8}{5}$ .

Therefore $ln (x_{1} x_{2}) = t_{1} + t_{2} = \frac{8}{5}$ , so $x_{1} x_{2} = e^{8/5}$ .

Concept behind this question

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