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4-2.Quadratic Equations and Inequations
hard
સમીકરણ $\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0$ ના ઉકેલની સંખ્યા મેળવો.
A$2$
B$4$
C$1$
D$3$
(JEE MAIN-2025)
Solution
$\text { Consider } \frac{1}{\sqrt{x}}=\alpha x>0$
$\left\{9 \alpha^2-9 \alpha+2\right\}\left\{2 \alpha^2-7 \alpha+3\right\}=0$
$(3 \alpha-2)(3 \alpha-1)(\alpha-3)(2 \alpha-1)=0$
$\alpha=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}, 3$
$x=9,4, \frac{9}{4}, \frac{1}{9}$
So, no. of solutions $=4$
$\left\{9 \alpha^2-9 \alpha+2\right\}\left\{2 \alpha^2-7 \alpha+3\right\}=0$
$(3 \alpha-2)(3 \alpha-1)(\alpha-3)(2 \alpha-1)=0$
$\alpha=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}, 3$
$x=9,4, \frac{9}{4}, \frac{1}{9}$
So, no. of solutions $=4$
Standard 11
Mathematics
