JEE Main 2025 Jan 29 Shift 1, Mathematics Q9: Some Other Forms of Equations

JEE Main2025Jan 29, Shift 1Mathematics

The number of solutions of the equation

$(\frac{9}{x} - \frac{9}{x} + 2) (\frac{2}{x} - \frac{7}{x} + 3) = 0$ is :

A
$2$
B
$4$
C
$1$
D
$3$

Solution

Set $α = \frac{1}{x}$ (with $x > 0$ ). Then $\frac{1}{x} = α^{2}$ , and the equation becomes

$(9 α^{2} - 9 α + 2) (2 α^{2} - 7 α + 3) = 0.$

Factor: $9 α^{2} - 9 α + 2 = (3 α - 1) (3 α - 2)$ and $2 α^{2} - 7 α + 3 = (2 α - 1) (α - 3) .$

The four roots are $α = \frac{1}{3}, \frac{2}{3}, \frac{1}{2}, 3.$ Each is positive, so each yields a valid $x = 1/ α^{2}$ : $x = 9, \frac{9}{4}, 4, \frac{1}{9} .$

Hence the equation has $4$ solutions.

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JEE Main 2025 Jan 29 Shift 1, Mathematics Q9: Some Other Forms of Equations

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