JEE Advanced 2022 Paper 2, Mathematics Section 1 Q4: Quadratic Equations

JEE Advanced2022Paper 2Mathematics Section 1

The product of all positive real values of $x$ satisfying the equation $x^{(16 (l o g_{5} x)^{3} - 68 l o g_{5} x)} = 5^{- 16}$ is _____.

Your answer

Solution

Take $lo g_{5}$ on both sides. With $t = lo g_{5} x$ :

$(16 t^{3} - 68 t) \cdot t = - 16 \Rightarrow 16 t^{4} - 68 t^{2} + 16 = 0 \Rightarrow 4 t^{4} - 17 t^{2} + 4 = 0$ .

Factor: $(4 t^{2} - 1) (t^{2} - 4) = 0$ , giving $t = \pm \frac{1}{2}, \pm 2$ .

The corresponding positive real roots are $x = 5^{1/2}, 5^{- 1/2}, 5^{2}, 5^{- 2}$ .

Their product is $5^{1/2 - 1/2 + 2 - 2} = 5^{0} = 1$ .

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