Question

If $x_1$ and $x_2$ are the roots of the equation $e^2.x^{\ln x}=x^3$ with $x_1>x_2$, then what is the relation between the roots?

• A. $x_1=2x_2$
• B. $x_1=(x_2{)}^2$
• C. $2x_1=(x_2{)}^2$
• D. $(x_1{)}^2=(x_2{)}^3$

Answer 1

The correct option is B $x_1=(x_2{)}^2$

Given, $e^2.x^{\ln x}=x^3$

$\ln (e^2.x^{\ln x})=3\ln x$

$\ln e^2+\ln \left(x^{\ln x}\right)=3\ln x$

$2+{\ln }^{2}x=3\ln x$

${\ln }^{2}x-3\ln x+2=0$

$(\ln x-2)(\ln x-1)=0$

$\ln x=2$ and $\ln x=1$

$x=e^2$ and $x=e$

As $x_1>x_2$

$x_1=e^2$ and $x_2=e$

Then $x_2^2=x_1$