Number of Solutions of Absolute Equation

CAT 2019 Slot 1 · Quantitative Ability · Medium · Quadratic equations

This is a medium Quantitative Ability question from the CAT 2019 Slot 1 paper. It tests Quadratic equations. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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The number of solutions of the equation $|x|(6x^2 + 1) = 5x^2$ is

Answer

Explanation

$x = 0$ is a solution. For $x \ne 0$ , divide by $x^2$ : $\frac{|x|(6x^2+1)}{x^2} = 5 \implies \frac{6x^2+1}{|x|} = 5$ . Let $t = |x| > 0$ . $\frac{6t^2 + 1}{t} = 5 \implies 6t^2 - 5t + 1 = 0 \implies (2t - 1)(3t - 1) = 0$ . $t = 1/2$ or $t = 1/3$ . Since $t = |x|$ , $x = \pm 1/2$ and $x = \pm 1/3$ . Total solutions $= 1 + 2 + 2 = 5$ .

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Number of Solutions of Absolute Equation

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